H2 Measurement 2012

7/25/2019 H2 Measurement 2012

1/21

1

1. MEASUREMENT

Content

SI units

Errors and uncertainties

Scalars and vectors

Learning Outcomes

Candidates should be able to:

(a) recall the following base quantities and their units: mass (kg), length (m),

time (s), current (A), temerature (!), amount of substance (mol)"

(b) e#ress derived units as roducts or quotients of the base units and use the

named units listed in $Summar% of !e% &uantities, S%mbols and 'nits as

aroriate"

(c) show an understanding of and use the conventions for labeling grah a#es

and table columns as set out in the ASE ublication SI Units, Signs, Symbols

and Abbreviations, e#cet where these have been suerseded b% Signs,

Symbols and Systematics (The ASE Companion to 16 1 Science, !"""#$

(d) use the following refi#es and their s%mbols to indicate decimal submultiles or multiles of both base and derived units: ico (), nano (n),

micro (), milli (m), centi (c), deci (d), kilo (k), mega (*), giga (+), tera ()"

(e) make reasonable estimates of h%sical quantities included within the

s%llabus"

(f) show an understanding of the distinction between s%stematic errors

(including -ero errors) and random errors"

(g) show an understanding of the distinction between recision and accurac%"

(h) assess the uncertaint% in a derived quantit% b% simle addition of actual,

fractional or ercentage uncertainties (a rigorous statistical treatment is not

required)"

(i) distinguish between scalar and vector quantities and give e#amles of each"

(.) add and subtract colanar vectors"

(k) reresent a vector as two erendicular comonents"

RAFFLES INSTITUTION PHYSICS DEPARTMENTYear 5


2/21

/ 0 1 2

unit

numerical magnitudeh%sical quantit%

2

1.1 S.I. Units

Physical Quantities he laws of 3h%sics are e#ressed as mathematical relationshis among h%sical

quantities and are verified through measurements of these quantities"

All h%sical quantities consist of a numerical magnitudeand a unit" E"g" in $aforce of five newtons4, $force4 is the h%sical quantit%, $five4 is the numerical

magnitude and $newtons4 is the unit"

he Summar% of !e% &uantities, S%mbols and 'nits used in the A5evel

e#amination is given in the s%llabus, which can be found in en 6ears Series or

online at '75: htt:88www"seab"gov"sg8a5evel89;?>@9;, the international scientific communit% has adoted a number ofconventions about h%sical quantities and their units" he Syst%me Internationale

d&Unit's (International S%stem of 'nits) is based on seven base quantities and

their corresonding units, called base nits"

!uantity "nit name "nit symbol

5ength metre m

*ass kilogram kg

ime second s

Current amere A

emerature kelvin !

Amount of substance mole mol

5uminous intensit%B candela cd

*out of s%llabus

Aendi# ; lists the definitions for these base units"


3/21

!eri"ed units )erived *antities+nits are obtained from the base quantities8units according to a

defining equation"

E"g" he defining equation for seed is

distanceseed

time

and hence, the unit of seed

is metreersecond (m s;

)"#Note the space brea$ beteen the m and the s%&. 'his is standard in separating

unit symbols in riting or in print. therise, e get ms%&, hich is an inverse

millisecond*

*ost derived units are given secial names for convenience"

!uantity +pressed in base units pecial name ymbol

Dolume m m m 0 m<

Delocit% m s 0 m s;

/orce kg (m s s) 0 kg m s9 newton 2

ork done kg m s9 m 0 kg m9s9 .oule F

Since derived units deend on base units, their si-e ma% change if these base units

become redefined or get ad.usted in value"

Gerived units are defined in a logical sequence"

In the stud% of electricit%, note how each h%sical quantit% is defined

from the revious quantit%"

!uantity"nit

(symbol)efinition

/ased

on

Current amere

(A)

the stead% current flowing in two straight,

infinitel% long and arallel conductors of circular

crosssection, laced one metre, aart in a

vacuum, which will roduce a force of 9 ;H2

acting on a metre length of conductor"

kg,

metre,

second

Charge coulomb

(C)

the amount of electrical charge that flows er

second through an% crosssectional area of a

conductor which carries a current of one amere"

amere,

second

3otential

difference

volt

(D)

the otential difference between two oints, when

one .oule of work is done (or one .oule of energ%

is e#ended) to bring one coulomb of charge from

one oint to the other "

.oule,

coulomb

7esistance ohm

()

the resistance of a circuit comonent, so that

when one amere is flowing through it, it

generates a otential dro of one volt across it"

amere,

volt

01!: E#lain wh% a volt should not be defined as the otential dro across a resistance

of one ohm when one amere of current is flowing through it"

1ns: he ohm is defined in terms of the amere and the volt" Since the ohm deends

on the si-e of the volt, it would be illogical to define the volt as the otential difference

across one ohm when one amere flows through it"

#$am%le


4/21

&

'omogeneity o(#quations

Ever% term on both sides of the equal sign of an equation should have the same

units, for the equation to be called homogeneos or dimensionally consistent" his

is .ust lain common sense, as when - . / 0, we e#ect all quantities,, .and

0 to reresent the same item"

#$am%le'esting s2 ut3 at4

'nit ofs0 m (metre)

'nit of t0 (unit of velocit%) (unit of time) 0 m s;s 0 m

'nit of at90 m s9s9 0 m

Equation is homogeneous"

#$am%le 'esting v2 u4 3 4as4

'nit of v0 m s;(metre er second)

'nit of 9 0 (unit of velocit%)90 (m sl) 9 0 m9s9

'nit of as90 m s9 m90 m


5/21

)

anal%sis"

!ecimalsubmulti%les andmulti%les

he following refi#es and their s%mbols can be used to indicate decimal sub

multiles or multiles of both base and derived units:

5refi ico nano micro milli centi deci kilo mega giga tera

ymbol n L m c d k * +

6ultiple ;;9 ;= ;> ;< ;9 ;; ;< ;> ;= ;;9

7.777789 ; can be epressed as 89. 477 ? can be epressed as &.==>4 6?

Standard (orm Standard form e#resses a number as N &7n where n is an integer, either

negative or ositive, and Nis an% number such that ;" M 2 M ="==="

#$am%le 7.777@> ; can be epressed as @.> &7%;

4=9777 m can be epressed as 4.=9

&7Am

Con"entions (orlabeling tablecolumns and gra%ha$es

All table columns and a#es of a grah must be labelled aroriatel%, al%ing

the correct use of the decimal submultiles8multiles and the standard form

format"

#$am%le In the table belo, columns 4 and > correctly tabulated the

values for

;

T but column > has a better presentation.

T8 ! ;;

8 !T

< ;; 8 ;: ! T

9H< ">


6/21

; 9


7/21

;

;

main scale

vernier scale

/

Current, Doltage *ultimeter B

B Geends on sensitivit% of instrument

0he ernier Calli%ers

It consists of a steel bar with a ermanent .aw and a sliding .aw" he movable .aw

carries a vernier scale that moves alongside the main scale" his instrument can

measure the internal and e#ternal dimensions of tubes, and the tail can measure the

deth of holes"

he vernier scale carries ten divisions that coincide with nine divisions of the main

scale" he whole idea is to have intervals of length on the vernier which are "= of

the millimetre interval on the main scale" he difference of ;" "= mm is called the

least count" All readings therefore become integral multiles of the least count"

he diagrams show the vernier scales when the instrument is closed and when it is

oen to measure an ob.ect4s diameter"

!escri%tion

Princi%le

#$am%le


8/21

1 ;

;

mm bserved Ceading 2 4.A3(&9

7.7&) 2 4.89

mm

Bength of obGect 2 Eorrected Ceading 2 4.89 7.7> 2 4.8 mm

#rrors In the rocess of taking measurements, we ma% encounter errors, sometimes

knowingl% and sometimes unknowingl%, that give rise to false readings"

Errors arise due to man% reasons:

the instruments ma% not be roerl% set u or calibrated,

the% ma% not be working roerl%,

the scales are misread,

the e#erimenter took down readings wrongl%,

disturbances ma% have taken lace"

he errors fall into two broad categories" he% are either systematic errors or

random errors"

andom #rrors

7andom errors roduce readings thatscatterabout a mean value" hese errors have an equal chance of

being ositive (making the readings too large) or negative (making readings too small)" he% can be

reduced b% taking more readings and averaging"

E#amles of random errors:

i" aralla# error

ii" fluctuation in the countrate of a radioactive deca%

Paralla$ error 3aralla# error takes lace when the line of sight of the e#erimenter is not

erendicular to the scale" he randomness in the error ma% occur when the

angle of the line of sight varies when reeated measurements are taken"

o reduce aralla# error, the e#erimenter should:

3lace an attached ointer or the ob.ect being measured as close to the

scale as ossible"

7eeat the measurement and find the mean of the reeated

measurements"


10/21

15

Count*rate o( aradioacti"e source

he e#ected value redicted for a articular longlived isotoe ma% be ;

counts er minute" 7eeated counts would reveal the data to scatteraround the

value of ; due to the random nature of radioactive deca%"

owever, the more counts the e#erimenter takes, the more accurate the average

becomes (i"e" closer to ;)" The random errors tend to cancel each other ot,

and the residal error is divided by the nmber o4 readings, so it gets shared ot

among many readings$ (se4l statement to remember5#

Systematic #rrors

A s%stematic error will result in all the readings taken differing from the true value b% a fi#ed ositive

amount (or negative amount)" A s%stematic error can be eliminated onl% if the source of the error is

known and accounted for" It cannot be eliminated b% averaging but by correct laboratory practice"

E#amles of s%stematic errors:

i" -ero error

ii" ersonal error of the observer, e"g" a mistimed actioniii" background radiation

6ero error An instrument is said to have a3ero error when the scale reading is non-ero

before an% measurement is taken" here aroriate, instruments should be

checked for -ero error and ad.ustment be made if ossible" he

resence8absence of -ero error should be recorded"

Stdents have a tendency to 4orget to record the absence o4 3ero error$

+is*timed 7ction Another e#amle of a s%stematic error is when a scientist kees doing a mis-

timed action" he scientist alwa%s starts his stowatch "1 s too late and stos it

on time" hatever time he records is alwa%s "1 s too small" e cannot correct

for this if he is not aware of what he has done" e will alwa%s carr% this

s%stematic error in all his time e#eriments"

'uman reaction time he dela% between the e#erimenter observing an event and starting a stowatch

is known as his reaction time" he reaction time of a normal human being is

between "9 s to "? s"

he effect of reaction time ma% be reduced b%

starting and stoing the stowatch to the same stimulus" /or e#amle,

use a 4idcial marerwhen measuring the eriod of a endulum" hee#erimenter is to react (start or sto) when the bob asses the marker"


11/21

Each reading taken is indicated b% one arrow"2

d 8 mm

;"= 9" 9"; 9"9 9"?9"


12/21

d 8 mm

;"= 9" 9"; 9"9 9"?9"


13/21

1

measurement is ritten as LDM 2 (47.> J 7.>) cm.

Signi(icant (igures

he number of significant figures in a result is siml% the number of figures that

are known with some degree o4 reliability"

#$am%le 7.477 g has > significant figures

A.787 &7; has significant figures

All uncertainties are rounded off to ; significant figure" he measured quantities

are then rounded off to the same d"" as the uncertainties"

eliability

7eliabilit% is a measure of confidence that can be laced in a set of

measurements" here are several wa%s to gauge reliabilit%:

Evaluate whether the data collected follows a articular trend as redictedb% a theor%" In articular, the scatter of the oints around the line of best

fit rovides evidence of reliabilit%"

3erform statistical anal%sis to obtain quantitative assessment of reliabilit%"

Evaluate the closeness of the relicates of the measurements"

/or simlicit%, a set of measurements is reliable if it is both accurate and recise"

Uncertainties o(deri"ed quantities he uncertainties of derived quantities (erimeter, area, volume, densit%, etc) ma%

be calculated using the following formulae:

;

"Addition ;0 a.J b0 Absolute error is

;0 a.J b0

9

"Subtraction ;0 a. b0 Absolute error is

;0 a.J b0

+

A

>

1,

Scalars - ectors All h%sical quantities can be divided intoscalarand vectorquantities"

A vectorquantit% has both a magnitude and a direction" Ascalarquantit% has a

magnitude onl%"

elow are e#amles of each t%e of quantit%:

calar !uantity ?ector !uantity

distance dislacement

mass force

seed velocit%

charge torque

temerature momentum

time acceleration

volume

energ%

e%resentation o( aector

hereas scalars are reresented onl% b% a number reresenting its magnitude, and

a unit, vectors are reresented b% a number, a unit and a direction"

A vector is denoted b%A , a or in most books A$

A vector is reresented b% an arrow:

the direction of the arrow reresents the direction of the vector,

the length of the arrow reresents the magnitude of the vector"

ector 7ddition 9 or more vectors ma% be summed u to form a resultant vector"

Consider the vector addition of two vectors A and> :

It can be seen that if A and > form the

sides of a arallelogram, then A >+ isthe diagonal of the arallelogram"

his is called the arallelogram method

of vector addition and it is equivalent to

the vector triangle"


17/21

>

A

>

A

A>

( )A >+ ur ur

A

>

v

(?1o

( )

= + %v

v (

1/

ector Subtraction

ere are two vectors, A and > " 7eversing them give A and >

If vector > is to be subtracted from vectorA , the resultant vector is A> "

A> can be rewritten as A J (> )" e can now use vector addition"

#$am%le 1n obGect as initially moving ith a constant speed of 47 m s-&

toards the east. 'hen it moved ith a constant speed of

&7 m s-&in the north-easterly direction. "sing vector analysis,

determine the change in velocity. #Note: the change in velocityis a vector quantity.*

he change in velocit% 0 ( )v (+

he magnitude is found using cosine rule"

9 9

9 9 o

;

9 cos

9: ;: 9 9: ;: cos ?1

;?"H m s

v ( v (v

= +

= +

= 'sing sine rule,

o

o

o

sin ?1 sin

;:sin ?1sin

;?"H?

9N"H

=

=

=

v v

he change of velocit% is ;?"H m s;in the direction of 9N"Ho2orth of est"


18/21

v

?1o

v(

sinA

AcosA

A

A

1

In the revious e#amle, the equation

change of velocit%, = v v (

can be rewritten as:

= + v ( v

ence, the vector diagram reresenting , vand vcan also be drawn as shown

below:

vcan be calculated using cosine rule, after which can be calculated using sine

rule"

'he problem can also be solved by draing a scale diagram.

esolution o(ectors

Dectors can be resolved into comonents in an% direction" 'suall%, the directions

chosen are the vertical and hori-ontal directions"

#$am%le 1 horse pulls on a rope that is attached to a barge, ith a forceof 87 N. 'he rope ma$es an angle of >7

oith the ban$s of the

river. etermine the magnitude of

(a) the force that pulls the barge forard and

(b) the force that pulls it sideays


19/21

mg

#

%

;9" 2

H" 2


20/21

25

7PP#:!I; 1 !e(initions o( S.I. Base Units

he metreis (9== H=9 ?1N);of the distance light travels in one second"

he $ilogramis equal to the mass of the International 3rotot%e kilogram (a latinumiridium c%linder) ket in

Sevres, 3aris"

he secondis defined in terms of = ;=9 > square metre of a

blackbod% at the temerature er square metre"

he kilogram is the onl% base unit defined b% a h%sical ob.ect" he rest of the base units are based on stable

roerties of the universe" /or e#amle, the metre is defined b% stating that the seed of light, a universal h%sical

constant, is e#actl% 9== H=9 ?1N metre er second" his h%sical definition allows scientists to reconstruct metre

standard an%where in the world without referring to a h%sical ob.ect ket in a vault somewhere"

7PP#:!I; 2 !eri"ation o( 9ormulae (or Calculating Uncertainties

Addition of &uantities

If;0 a.J b0

a(..) J b(0 0) 0 (a. J b0) (a.J b0)

ence ; 0 a.J b0

Subtraction of &uantities

If;0 a.b0

a(..) b(0 0) 0 (a. b0) a. b0

-(a. b0) (a.J b0)

ence ; 0 a.J b0

3roduct of &uantities

If ;0.m0n , (..)m(0 0)n-

; " ;

m n

m n. 0. 0

. 0

0

; ;

m n

m n . 0. 0

. 0

E#anding binomiall% taking first order aro#imation,

(..)

m

(0 0)

n

; ; =

m n . 0. 0 m n

. 0

;;

;;


21/21

21

; =

m n . 0 . 0

. 0 m n mn. 0 . 0

Tlast term ignored for small . ? 0U

= +

m n m n. 0. 0 m n . 0

. 0

Since

m n. 0 . 0; m n . 0 m n ;. 0 . 0

= + = +

herefore,

; . 0m n

; . 0

= +

;;

Date post:	24-Feb-2018
Category:	Documents
Upload:	ronnie-quek
View:	216 times
Download:	0 times

H2 Measurement 2012

Documents