Oscillations - P2, week 5

7/29/2019 Oscillations - P2, week 5

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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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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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6/18

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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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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10/18

ForcesandSHM

Mo+onofobjectsisdeterminedbytheforcesac+ngonthem ItisthesamewithSHM

ForSHMwehavea=-2s Newtons2ndLaw:F=maF=-m2s

Fisthenetforceontheobject misthemassoftheobject

ThenetforceisALWAYStryingtorestoretheobjecttoequilibrium

Thenetforceisstrongerthefurtherawaytheobjectisfromtheequilibriumposi+on

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11/18

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

2 15


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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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Oscillations - P2, week 5

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