Download - H2 Measurement 2012
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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
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/ 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 $newtons4 is the unit"
he Summar% of !e% &uantities, S%mbols and 'nits used in the A5evel
e#amination is given in the s%llabus, which can be found in en 6ears Series or
online at '75: htt:88www"seab"gov"sg8a5evel89;?>@9;, the international scientific communit% has adoted a number ofconventions about h%sical quantities and their units" he Syst%me Internationale
d&Unit's (International S%stem of 'nits) is based on seven base quantities and
their corresonding units, called base nits"
!uantity "nit name "nit symbol
5ength metre m
*ass kilogram kg
ime second s
Current amere A
emerature kelvin !
Amount of substance mole mol
5uminous intensit%B candela cd
*out of s%llabus
Aendi# ; lists the definitions for these base units"
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!eri"ed units )erived *antities+nits are obtained from the base quantities8units according to a
defining equation"
E"g" he defining equation for seed is
distanceseed
time
and hence, the unit of seed
is metreersecond (m s;
)"#Note the space brea$ 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"
#$am%le
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'omogeneity o(#quations
Ever% term on both sides of the equal sign of an equation should have the same
units, for the equation to be called homogeneos or dimensionally consistent" his
is .ust lain common sense, as when - . / 0, we e#ect all quantities,, .and
0 to reresent the same item"
#$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"
#$am%le 'esting v2 u4 3 4as4
'nit of v0 m s;(metre er second)
'nit of 9 0 (unit of velocit%)90 (m sl) 9 0 m9s9
'nit of as90 m s9 m90 m
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anal%sis"
!ecimalsubmulti%les andmulti%les
he following refi#es and their s%mbols can be used to indicate decimal sub
multiles or multiles of both base and derived units:
5refi ico nano micro milli centi deci kilo mega giga tera
ymbol n L m c d k * +
6ultiple ;;9 ;= ;> ;< ;9 ;; ;< ;> ;= ;;9
7.777789 ; can be epressed as 89. 477 ? can be epressed as &.==>4 6?
Standard (orm Standard form e#resses a number as N &7n where n is an integer, either
negative or ositive, and Nis an% number such that ;" M 2 M ="==="
#$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< ">
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; 9
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;
;
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
#$am%le
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1 ;
;
mm bserved Ceading 2 4.A3(&9
7.7&) 2 4.89
mm
Bength of obGect 2 Eorrected Ceading 2 4.89 7.7> 2 4.8 mm
#rrors In the rocess of taking measurements, we ma% encounter errors, sometimes
knowingl% and sometimes unknowingl%, that give rise to false readings"
Errors arise due to man% reasons:
the instruments ma% not be roerl% set u or calibrated,
the% ma% not be working roerl%,
the scales are misread,
the e#erimenter took down readings wrongl%,
disturbances ma% have taken lace"
he errors fall into two broad categories" he% are either systematic errors or
random errors"
andom #rrors
7andom errors roduce readings thatscatterabout a mean value" hese errors have an equal chance of
being ositive (making the readings too large) or negative (making readings too small)" he% can be
reduced b% taking more readings and averaging"
E#amles of random errors:
i" aralla# error
ii" fluctuation in the countrate of a radioactive deca%
Paralla$ 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"
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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"
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Each reading taken is indicated b% one arrow"2
d 8 mm
;"= 9" 9"; 9"9 9"?9"
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d 8 mm
;"= 9" 9"; 9"9 9"?9"
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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"
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>
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"
#$am%le 1n obGect as initially moving ith a constant speed of 47 m s-&
toards the east. 'hen it moved ith a constant speed of
&7 m s-&in the north-easterly direction. "sing vector analysis,
determine the change in velocity. #Note: the change in velocityis a vector quantity.*
he change in velocit% 0 ( )v (+
he magnitude is found using cosine rule"
9 9
9 9 o
;
9 cos
9: ;: 9 9: ;: cos ?1
;?"H m s
v ( v (v
= +
= +
= 'sing sine rule,
o
o
o
sin ?1 sin
;:sin ?1sin
;?"H?
9N"H
=
=
=
v v
he change of velocit% is ;?"H m s;in the direction of 9N"Ho2orth of est"
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v
?1o
v(
sinA
AcosA
A
A
1
In the revious e#amle, the equation
change of velocit%, = v v (
can be rewritten as:
= + v ( v
ence, the vector diagram reresenting , vand vcan also be drawn as shown
below:
vcan be calculated using cosine rule, after which can be calculated using sine
rule"
'he problem can also be solved by draing a scale diagram.
esolution o(ectors
Dectors can be resolved into comonents in an% direction" 'suall%, the directions
chosen are the vertical and hori-ontal directions"
#$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
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mg
#
%
;9" 2
H" 2
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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
;;
;;
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; =
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
= +
;;