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PhysicsAS-Unit2-Mechanics,MaterialsandWaves-RevisionNotesMechanics

ScalarandVectorQuantities DefinitionAVectorQuantityhasmagnitudeaswellasdirectionwhileascalar

quantityonlyhadmagnitude

ScalarDistance,Speed,Mass VectorsDisplacement,Velocity,Acceleration

Resultant

Pythagoras- a = b2+ c

2

Note:whenyouknowtheresultantanddrawingthehorizontalandverticalcomponentdraw

bothofthecomponentsfromthesamepointthattheresultantstartsfrom(resultantshouldbe

in-between2components)

ResolutionOfVectors Resolvesvectorintohorizontalandverticalcomponents.

x= acos y = asin

BalancedForces Ifanobjectisinequilibriumthetwoforcesactinguponapointareequalandoppositeto

eachotherandthesumoftheanticlockwisemoments=thesumoftheclockwise

moments.Noresultantforceacting.

Equilibrium3forces2oftheforces(resultant)areequalandoppositetothethirdforce,thereisnoresultantforceacting.Canalsobedrawninaclosedtriangle.

Tocalculateandunknownforceorforcesresolveeachhorizontallyandverticallyandsolve,

ifobjectisinequilibriumthereisnoresultantforce!

a b

a+b

ParallelogramMethod

a

x

y

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Moments DefinitionThemomentofaforceaboutapointisequaltotheforcemultipliedby

theperpendiculardistancebetweenthelineofaction(offorce)andthepivot

(placetakingmomentabout)

Theprincipleofmomentsstatesthatforanobjecttobeinequilibriumthesumoftheanticlockwisemomentsmustequalthesumoftheclockwisemomentsatanypoint.

TwoSupportProblems Ifcenterofmassofbeamismidwaybetweenbothsupportstheforceeachsupport

exertsisequal.

Ifnot:o TakemomentsaboutS1tocalculateS2toworkoutS1orviceversa

Couples DefinitionAcoupleispairofequalandoppositeparallelforcesactingonabody

butnotatthesamepoint.

TheturningeffectofacoupleofequalandoppositeforcesisequaltoOneoftheforcesmultipliedbytheperpendiculardistancebetweenthetwoforces.

MotionAlongaStraightLine

DisplacementTimeGraphs

Gradient=VelocityVelocityTimeGraphs

Gradient=Acceleration AreaUnderneath=Displacement(ifvelocityisalways+)

ConstantAccelerationFormulas

S= ut+ 12

at2

S= 12(u+ v)t

v = u+ at v2 = u2 +2as

WhereV=finalvelocity,S=displacement,U=initialvelocity,T=time

RememberAccelerationisms-2

Whencalculatingdistancecoveredwhensomethingisacceleratingandstartingvelocityisnot0useformulas!!

IFEVERASKSFORDECELERATIONNEVERUSEAMINUSSIGNISITISSLOWINGDOWN!

ProjectileMotion

Accelerationduetogravity=9.81ms-2 Resolvehorizontallyandvertically Gravityonlyeffectstheverticalcomponentofmotion

NewtonsFirstLawOfMotion Objectsstayatrestorremaininuniformmotionunlessacteduponmyanexternal

force

Sobasicallyifanobjectismovingandnothingisactinguponit(e.gairresistance)theobjectwillcarryongoingforever,thishappensinspaceasnoairresistance

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NewtonsSecondLawofMotion Theforceappliestoanobjectisproportionaltothemassandtheaccelerationof

theobjectasaresultofthebeingapplied

F=ma If2forcesareactingupontheobjectitistheresultantforcethatisusedinthisequation-

F1F

2ifF

1isbiggerthanF

2

Witharocketthatistravellingdirectlyupwardstheobjecthastoovercomeitsownweight(mg )inordertoacceleratesoFmg =ma

Withalifto Ifmovingataconstantspeedupwards F=mg o Ifmovingupwardsandaccelerating F=mg+ma >mg o Ifliftmovingupanddecelerating F=mg+ma mg

o Ifmovingdownwardsanddecelerating F=mg+ma

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MeasuredinJoules

KineticEnergy

Ek =1

2mv

2

GravitationalPotentialEnergy

Eg =mgh Asanobjectofmassmisreleasedabovetheground,itgravitationalpotentialenergyis

convertedintokineticenergyastheheightabovegrounddecreasesas Eg =mgh ,inreality

thisisnotstrictlytrueassomegravitationalpotentialisconvertedintoworkagainstAir

resistanceandsomelostasheatandsound.

PendulumBob

ifreleasedfromheight hithenatanypointwheretheheightish

vthespeedofthebob

issuchthatKineticenergyatthatpoint=lossingravitationalpotential(ignoringalloverexternalforces)

1

2mv

2=mg(hi hf )

Power PowerRateoftransferofenergymeasuredinWatts

p =E

tp =

W

tp = fv

WhereE=Energy,W=Workandt=time Power=forcexvelocityasitstherateworkisdonepersecondasitsforcemultipliedby

distancemovedeachsecond

Efficiency

=

=

Sobasicallyefficiencyisoutputpoweroverinputpower

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Materials

Density DensityAmaterialsmassperunitvolume

p =m

v

Massofalloy,m = pava + pbvb andthedensityofthatalloy p = mv=

pava + pbvb

m

Springs

HookesLaw

HookesLawTheextensionofaspringisproportionaltotheforceappliedtoitaslongasthelimitofproportionalityisnotexceeded

F= kl (wherekisthespringconstant)

ElasticLimitMaximumstressthatcanbeappliedtoamaterialwithoutelasticbehavioroccurringdeformation

Note:ElasticlimitmaybedifferenttoLimitofProportionality

YieldPointApointisreachedatwhichnoticeablylargerchangeinlengthduetoforce BreakingStress/UltimateTensileStress moststressamaterialcanwithhold

withoutbreaking

BrittlenessMaterialsthatcantextendwithoutbreakingaresaidtobebrittlehoweverbrittlematerialsarestrongincompressionandusedinbuildingwherecompressionis

constant.

Springsinparallel- K= K1 +K2 wherekistheeffectivespringconstant Springsinseries- 1

K=

1

K1

+

1

K2

Inordertocompare2differentmaterialselasticpropertiesuseyoungsmodulus

YoungsModulus

YoungsModulusStiffnessconstantofmaterial TensileStress=forceperunitarea= f

ameasuredinNm-2

TensileStrain=ratiooforiginallengthandextension= LL

NOUNITS

ThisisdeltaLasinchangeinLsofinal-initial E=

=

Stress

StrainYoungsmodulushasunitsNm-2orPascal(Pa)

E=FL

AL

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ExperimentaldeterminationofYoungsModulusOfaMaterial

Twoidenticalwiresarefixedinparallel Bothwiresinitiallyloadedtoremovekinks/strengthenknots Micrometeradjustedtomakespritlevelhorizontal Initialmicrometerreadingtaken Meterrulermeasureoriginallengthofwirel(oneunder

test) Secondmicrometerusedtomeasurediameterofwirein

severalplacestoimproveaccuracy

Calculateareabyusing r2 (halfdtogiver) Testwireloadedwithmassandmicrometeradjustedtolevel

spritlevel

Micrometerreadingtakenandextensioncalculatedbyfinalmicrometerinitial

Furtherloadsareaddedandrepeateduntilrangeobtained Unloadsecondset Graphofforceagainstextension Gradientcalculatedusingbigtriangle

y2 y

1

x2 x

1

Asgradient= FL

andY =FL

ALYcanbecalculatedbygradientmultipliedby

y =GradientL

A

AccuracyImproved

Longthinwiregiveslargeextensionperunitforcethereforpercentageuncertaintydecreases

Controlwireusedsotemperaturechangesdonotimpactresults Measurediameterinseveralplacestogiveaverage Largetriangle/rangeusedtocalculategradient

LoadingandUnloading

IfelasticlimitisreachedtheunloadingcurvewillnothavesamevalueasloadingcurveatF=0ithasbeenmisshapedmetalextended

= Asloadingarea=workdoneandunloadingarea=energygivenback Rememberwhencountingsquarestocalculateworkdonethateachsquareisworth

whateverthesquarerepresentsverticallymultipliedbywhateveritrepresents

horizontally

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Waves

ProgressiveWavesWaveswhoseoscillationstravelanddonotstayaboutafixedpoint,thistypeofwavestransfersenergy

FrequencyNumberofwavecyclesthatoccurinonesecondmeasuredinHz Amplitude-Maximumdisplacementofavibratingparticlefromequilibrium WavelengthDistancebetween2adjacentpoints/particlesinphaseinawave Periodtimeforonecompletewavetopassafixedpointinspace

P =1

f

TransverseWavesWaveswhoseoscillations/vibrationsareperpendiculartothedirectionoftravel(lightetc.)

LongitudinalWaves Waveswhoseoscillations/vibrationareinthesamedirectionoftravel(sound)Note:Soundwavespropagateasaaseriesofcompressionsandrarefactions

WaveSpeedSpeedofthewavesisequaltodistancetraveledbywaveinonecycledividedbytimetakenforonecycle

C=

1f

Therefor c = f

Path/PhaseDifference

2pointsareinphaseiftheyareawholewavelengthapart(maxdisplacementatsametime)-thereoscillationsareintimewitheachother

2PointsareinAntiphaseiftheyarehalfawavelengthapart(onemaxdisplacementwhileotherexperiencesmindisplacement)

PhaseDifference

= 2x1 x

2

"

#$

%

&' (inradians)OR= 360

x1 x

2

"

#$

%

&' indegrees

note:x1-x2isthedistancebetweenthetwopoints

PathDifference

Between2differentsourcesthephasedifferencebetweenthe2wavesthentheyconvergeis:

= 360S1P S

2P

"

#$

%

&'

note:whereS1Pisthedistancefromsource1tothepoint

wherethe2wavesconverge

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Polarization

PolarizationWhenoscillationsoftheelectricfieldofanE.Mwavearerestrictedtoonlyoneplane/directionthewaveissaidtobepolarized

Application Polaroid sunglasses reduce glare from water as the light is partiallypolarizedwhen reflected offwater thus theintensityof the light can bereduced This

reducesglare

Reflection Angleofincidenceisthesameasangleofreflection

i = r

StationaryWaves

Superposition

When2similarwavesofsimilarfrequencymeettheirresultantdependsupontheiramplitudeandtheirrelativephasedifference

o Ifnophasedifferenceanglethenconstructiveinterferenceoccurso If180degreesdifferenceanglethendestructiveinterferenceoccursandwaves

canceloutatthatpoint

PrincipleOfSuperpositionStatesthattheresultantdisplacementcausedby2wavesarrivingatapointisthevectorsumofthe2displacementscausedbyeachwavesatthat

instant

StationaryWavesFormation

Formedwhen2continuouswavestravellinginoppositedirectionsofsamefrequencysuperimposeto

formnodisplacement(nodes-completedestructive)

andpointsofmaxdisplacement(antinodescomplete

constructiveinterference) StationaryWaveFixedpatternofvibrationwhere

noenergyistransferredalongthewave

Nodeparticle/pointwithzerodisplacementnoamplitude

AntinodePoint/particlewithmaxdisplacement/amplitude

Thephasedifferencebetween2particles/pointsonstandingwave(thatisthedifferencewhenoneisat

maxdisplacementandotheronealsois,indegrees)is

zeroifthepoints/particlesarebetweenadjacentnodesorseparatedbyevennumberofnodesor180

degreesiftheyareseparatedbyanoddnumberof

nodes.

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FundamentalFrequencies

Distancebetween2AdjacentNodes= 2

(asfrom

nodetonode=halfawavelengthofoneofthe

continuouswavesthatformedstationarywaveseen

indiagram) Asseenindiagramthefirstfundamentalthatiswhen

thereisonestandingwave(1anti-node1node)the

lengthishalfofthewavelength

= 2L

Using c = f

Wefind: f0=

c

2l

(wherecisspeedofpropagationofwaves)

Henceatthesecondharmonicorfirstovertone := L o Hence: f

1=

c

l

Furthermoreatthethirdharmonicorsecondovertone: = 23L

o Hence: f2=

c

23

l

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Refraction RefractionChangeofdirectionofawavewhenitpropagatesthroughadifferent

medium

Speedoflightinairis3.00106 howeverwhenitentersmoreopticallydensemediumitreducesspeed

Sowhenlightentersamoreopticallydensemediumitsspeeddecreasesanditsdirectionchangesmoretowardsthenormal As c = f andthespeedoflightisreducedinmoreopticallydensemediumthe

wavelengthoflightchanges(makessensereallythefrequencycantsuddenlychange!)AbsoluteRefractiveindex

=

note:asthespeedofdifferentfrequencyoflightisnotthesameinagivenmedium(duetothefactthe

propagationdirectionchangestoadifferentdegreeforeachfrequency)togiveabsoluterefractiveindex

Yellowlightfromasodiumbulbisusedasastandard

SometypicalR.Is

Diamond2.4

Perspex1.5 Air1

Ice1.31

RefractiveIndexbetween2materials

=

!!

Iflightfromonemedium1tomedium2isNthenlightfrom2to1is1/NSnellsLaw

n1sin

1= n

2sin

2

OR

sin1

sin2

=

n2

n1

=

c1

c2

=1

2

Notewherenistheabsoluterefractiveindexofacertainmedium

and1istheangleofincidenceand2istheangleofrefraction

TotalInternalReflection

Ifangleofincidenceissolargeandthelightismovingfromamoredensetolessdensemedium(hencepropagationdirectionchangingawayfromnormal)theangleof

incidencewillbelargerthan90andsubsequentlythelightdoesnotexitbutisreflected

backintomedium

Theanglethatthishappensatthatistheangleatwhichlightisreflectedat90tothenormal(

2=90)iscalledthecriticalangleandusually

denotedby i

sini

sin90=

n1

n2

andassin90=1:sin i =

n2

n1

Thisishowfiberopticcableswork:

CladdingInFiberOpticCables o

Improvetensilestrengthofthecableasthecorehastobeverythin.o Increasethecriticalangleneededfortotalinternalreflection:hencereducing

themultipathdispersion(mergingofsignalsoflightdownacable)

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o Stopscorebeingscratchedasscratchescauselighttobedispersedifithitsthescratch

o Improvessecurityaswithoutcladdinglightwouldbeabletotravelbetweenfibers,aswithcladdingtotalinternalreflectionoccursandlightcannottravelbetweenfibershoweverwithoutcladdinglightwouldbefreetotravelbetween

mediansofsimilaropticaldensity(coretocore)

UsedinInternetbroadbandtodeliverfastinternetconnectiontohouseholds.

Interference When2similarwavesmeetatapoint,bytheprincipleofsuperpositionthecombined

displacementisfoundbythevectorsumofthe2displacementsofeachwaveatthat

point,iftheyarriveexactlyinphasethewaveswillconstructivelyinterferetoforma

doubleheightwave,ifexactlyoutofphase(180degrees)theywilladdtogetherdestructivelyandcancel

Inordertoviewasteadyinterferencepatternthewavesourceshavetobecoherent CoherentConstantphaserelationshipandsamefrequency Thepatternweseewilldependuponthephasedifferenceofthe2sourceswhentheyhit

thatpoint

AbrightfringeWhenthewavesconstructivelyinterfere,occurswhenpathdifference= n

AdarkfringeWhenthewavesdestructivelyinterfere,occurswhenpathdifference= (n+

1

2)asthewaveshavetobe180degreesoutofphase

Two-Slitinterferencepatterns(YoungsSlits)

Singleslitdiffractionspreadsoutonewaveandcausesittoactas2coherentsourcesfor2furtherslits.

Showedevidenceforwavetheoryoflight Lightbandsoccurwheneverthepathdifference

betweenthelightwavesareawholenumberof

wavelengths(anddarkwhenhalfnumber) Thedistancebetweentwosuccessivemaximaw

dependsupon:

o Thedistancebetweenthe2slits;increasingSmakesfringesclosertogether(orwsmaller)

o Thewavelengthoflight,Wissmalleratsmallerwavelengthso DistanceDbetweentheslitsandthescreen,ifDincreaseWincreases.

w =D

s Alsocanuselaser

o Laserlightmonochromatico LaserlightIscoherent(constantphaserelationshipsameHz)o Highlydirectionalverylittledivergence

PatternMaximaSimilarintensityandsamewidthascentralfringe Diffraction

WhenwavespassthroughagaporaroundanobstaclethewavesspreadoutthisiscalledDiffraction

Whentheslitwidthisthesameasthewavelengthperfectdiffractionoccurs Whenslitislargethanwavelengthlessdiffractionoccurs

SingleSlitDiffractionPattern

Centralmaximumistwiceaswideastheothers

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Restofthefringesdecreaseslightlyinintensityaswayfromthemiddle(samewidth)

Maximaoccurswithconstructiveinterferenceandzerointensitywithdestructiveinterference

Ifgapgetssmallerofdistancebetweenscreenandslitincreasessodoeswidthofmaxima

Centralmaximagetslargerifwavelengthislongerorthegapissmaller

DiffractionGrating

Seriesofuniformnarrowslitsinparallel dsin= n

Whered=thedistancefromcentertocenterofadjacentslits(N=1/dtoworkoutdifyouonlyhavenumberofslitsinsayameter) istheangleoftheorderyouaretryingto

calculate,nistheordernumbertryingtocalculate12or3etc.

Fractionsofadegreeusuallyexpressedinminutes Tofindmaxnumberofordersubstitute for90(thereforsin=1)anduseequation Application:Spectrometer

o Usescollimatortoproduceparallellightandthenusesdiffractiongratingtoproducespectrumpattern

o Usesamplelightfromexcitinganatomtoproducepatternandanalysiscanbedoneonthis12

o AlwaysawhitelineattheZeroorderasallwavelengthsarriveinphaseatthispoint

o Theshorterthewavelengththeshortertheangleo Hencepatternscanbecomplicatedfirstorderredlightlinecanbecloseto

secondorderblue