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Oscillatingsystems
Anykindofmo+onthatrepeatsitselfperiodicallyaboutameanposi+onissaidtobeoscillatory
Examples
Thiskindofmo+oniscalledperiodicmo*on 1
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f =1
T
Afewkeyconcepts
PeriodT:The+meittakestocompleteonecycleoftheoscilla+on
Unit:Second Frequencyf:Thenumberofcyclescompletedpersecond
Unit:Hertz(Hz)=1cyclepersecond
AmplitudeA:Themaximumdisplacementfromtheequilibriumposi+on
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Circularmotion
Objectmovesinacirclearoundafixedpointwithuniformspeedv
In+met,theobjectmovesthroughanangle
Angularspeed=/t(radianspersecond) Timetocomplete1circle:periodT=2/ Rela+onshipbetweenspeedandangularvelocity:v=r
v
v
r
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SimpleHarmonicMotion(SHM)
Isaspecialkindofoscilla+ngmo+on Isassociatedwithcircularmo+on Examples:
massbobbingupanddownonaspring Pendulumswingingto-and-fro
Whatcanwesayabouttheposi+on,speed,displacementofanobjectinSHM?
Displacementchangeswith+me Thefurtherawaytheobjectisfromtheequilibriumposi+ontheloweritsspeed
Thespeedis0atmaximumdisplacement Atmaximumdisplacementtheobjectchangesdirec+on 4
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SimpleHarmonicMotion(SHM)
rrcos
Displacement
Velocity
Accelera+onv =
ds
dt= r
d
dtcos(t) = rsin(t)
s = rcos= rcos t( )
a =dv
dt= r
d
dtsin(t) = r
2cos(t)=
2s
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Alternativeexpressionforthe
velocityinSHM Fromtrigonometry:sin2(x)+cos2(x)=1
s = rcos(t) cos(t)=s
r
cos2(t) =
s2
r2
v = rsin(t) sin(t) = v
r
sin2(t) =
v2
r2
2
so
v2
r2
2+
s2
r2=1 v
2+ s
2
2= r
2
2 v = r
2 s
2
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Example
Amassishungatthebo]omofaspring.Thetopendofthe
springisfixedtotheceiling.Wepullthemassdownandletit
oscillatewithSHM.Theamplitudeoftheoscilla+onismeasured
tobe40mmandtheperiodis+medtobe0.35s.Whatisthe
speedofthemasswhenthedisplacementisa) Zerob) 20mm
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SimpleHarmonicMotion(SHM)
Remember:
Iftheaccelera+onofanobjectis-2stheobjectismovingwith
SHM
TheperiodTofthemo+onisthe+metakentocompleteoneto-
and-fromo+onorcycle
Thefrequencyf=1/TandismeasuredinHz
Theamplitude,r,isthemaximumdistanceeithersideof
theposi+onofequilibrium
s
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Example
Asteelstrip,lampedatoneend,vibrateswithafrequencyof
20Hzandanamplitudeof5mmatthefreeend,whereasmall
massof2gisposi+oned.Find
a) Thevelocityoftheendwhenpassingthroughtheequilibriumposi+on
b) Theaccelera+onatmaximumdisplacementc) ThemaximumK.E.ofthemass
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ForcesandSHM
Mo+onofobjectsisdeterminedbytheforcesac+ngonthem ItisthesamewithSHM
ForSHMwehavea=-2s Newtons2ndLaw:F=maF=-m2s
Fisthenetforceontheobject misthemassoftheobject
ThenetforceisALWAYStryingtorestoretheobjecttoequilibrium
Thenetforceisstrongerthefurtherawaytheobjectisfromtheequilibriumposi+on
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Springandmass
A:unloadedspringequilibriumposi+on
B:springloadedwithmassm HookesLaw:theextensionofaspringisindirectpropor+onwiththeloadappliedtoit
C:masspulleddownfurtherbyadistances,nolongerat
equilibrium
h
x0
A B C
s
Tension
Weight
Weight
Tension
F= k(x0+ s)+mg =ma
a = k
ms =
2s
T=2
= 2m
k
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Example
Aspringofnaturallength200mmishungfromtheceiling.We
thenhangamassof0.2kgfromitanditslengthbecomes
260mmatthenewequilibriumposi+on.(assumeg=9.8m/s2)
Calculate
a)thespringconstantb)theperiodoftheoscilla+onsifthespringispulleddownabit
furtherandreleased
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SimplePendulum
Non-elas+cthread:HookesLawdoesnotapply
Atequilibrium:Tension=mg Tensionalwaysperpendiculartothepathsocannotprovideaccelera+on
alongpath
Whendisplacedbyangle Paralleltopath Perpendiculartopath
L
Weight
mgcos
mgsin
mgsin=ma a =gsins
Forsmallangles:sin=ands=L
a=
g=
g
L s=
2
sSo (SHM)
FT mgcos=mv2
L
T=2
= 2L
gPeriod
TensionFT
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Example
Asmallmassof20goscillatesasasimplependulum,with
amplitude5cmandperiod2seconds.Findthespeedofthe
massandthetensioninthesuppor+ngthreadwhenthespeed
ofthemassisatmaximum.Assumeg=9.8m/s2
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K.E.andP.E.ofoscillating
systems Forundampedoscilla+ons:K.E.P.E.=constant
Averageforcerequiredtomovefromequilibriumtodistances
Workdoneinmovingmasstos TotalEnergy
v = r2 s
2
K.E. =1
2mv
2=
1
2m
2(r
2 s
2) since
F=1
2m
2s
W = Fs =1
2
m2s2= P.E.
E= P.E.+K.E. =1
2m
2s2+
1
2m
2(r
2 s
2) =
1
2m
2r
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K.E.andP.E.ofoscillating
systems
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Example
Amassof4kgisa]achedtoaspringandundergoesSHMwitha
periodof0.35s.Thetotalenergyofthesystemis2.5J.Es+mate
a) Thespringconstantb) Theamplitudeofthemo+on
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ToDo
Readsec+on5fromthebook[p.69-p.82Oscilla+ons HomeworkAssignmentwk5:ques+ons5.1,5.3,5.5,5.9,5.10,
5.13
Handitinnolaterthan4:00pmnextWednesday-LATEWORKWILLNOTBEACCEPTED
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