Download PHYS101

IntroductoryPhysics
PHYS101
Dr RichardH.CyburtOfficeHours
TRF9:30-11:00am
AssistantProfessorofPhysics
F12:30-2:00pm
Myoffice:402cintheScienceBuilding
Myphone:(304)384-6006
Meetingsmayalsobearrangedatothertimes,
byappointment
Myemail:[email protected]
Inpersonoremailisthebestwaytogetahold
Checkmyscheduleonmyofficedoor.
ofme.
PHYS101
PHYS101:IntroductoryPhysics
400
Lecture:8:00-9:15am,TRScienceBuilding
Lab1:3:00-4:50pm,FScienceBuilding304
Lab2:1:30-3:20pm,MScienceBuilding304
Lab3:3:30-5:20pm,MScienceBuilding304
Lab20:6:00-7:50pm,MScienceBuilding304
PHYS101
MasteringPhysicsOnline
GotoHYPERLINK"http://www.masteringphysics.com."www.masteringphysics.com.
◦ UnderRegisterNow,selectStudent.
◦ Confirmyouhavetheinformationneeded,thenselectOK!Registernow.
RCYBURTPHYS101),andchooseContinue.
◦ Enteryourinstructor’sCourseID(
◦ EnteryourexistingPearsonaccountusername andpassword andselectSignin.
◦ YouhaveanaccountifyouhaveeverusedaPearsonMyLab &Masteringproduct,suchasMyMathLab,MyITLab,MySpanishLab,or
MasteringChemistry.
◦ Ifyoudon’thaveanaccount,select Create andcompletetherequiredfields.
◦ Selectanaccessoption.
◦ Entertheaccesscodethatcamewithyourtextbookorwaspurchasedseparatelyfromthebookstore.
PHYS101
FinalExam
Tuesday,Dec6:9:00-11:15amRoom400
AllowedFullsheetofpaperwithformulae
andtrigcalculator(nocellphonesorotherdevices)
50%ofexamonnewmaterialChapters12-15:ReviewFri,Dec2.7-9pm,S300
50%ofexamonoldmaterialChapters0-11:ReviewMon,Nov28.7-9pm,S300
Make-UpLabsMon,Nov28&Fri,Dec2.
PHYS101
IntroductoryPhysics
PHYS101
DouglasAdams
Hitchhiker’sGuidetotheGalaxy
PHYS101
You’realreadyknowphysics!
Youjustdon’tnecessarilyknowtheterminologyand
languageweuse!!!
PhysicsofNASCAR
PhysicsofAngerBirds
PHYS101
Inclass!!
PHYS101
Thislecturewillhelpyouunderstand:
Equilibrium&Oscillation
LinearRestoringForces&SimpleHarmonicMotion
DescribingSimpleHarmonicMotion
EnergyinSimpleHarmonicMotion
PendulumMotion
PHYS101
Section14.1Equilibrium
andOscillation
©2015PearsonEducation,Inc.
EquilibriumandOscillation
Amarblethatisfreetoroll
insideasphericalbowlhas
anequilibriumposition at
thebottomofthebowl
whereitwillrestwithno
netforceonit.
Ifpushedawayfrom
equilibrium,themarble’s
weightleadstoanetforce
towardtheequilibriumposition.Thisforceistherestoringforce.
©2015PearsonEducation,Inc.
EquilibriumandOscillation
Whenthemarbleisreleased
fromtheside,itdoesnot
stopatthebottomofthe
bowl;itrollsupanddown
eachsideofthebowl,
movingthroughthe
equilibriumposition.
Thisrepetitivemotioniscalledoscillation.
Anyoscillationischaracterizedbyaperiod andfrequency.
©2015PearsonEducation,Inc.
FrequencyandPeriod
Foranoscillation,thetimeto
completeonefullcycleis
calledtheperiod(T) ofthe
oscillation.
Thenumberofcyclesper
secondiscalledthefrequency
(f )oftheoscillation.
Theunitsoffrequencyarehertz (Hz),or1s–1.
©2015PearsonEducation,Inc.
QuickCheck14.3
Atypicalearthquakeproducesverticaloscillationsoftheearth.Supposea
particularquakeoscillatesthegroundatafrequencyof0.15Hz.Astheearth
movesupanddown,whattimeelapsesbetweenthehighestpointofthe
motionandthelowestpoint?
◦
◦
◦
◦
1s
3.3s
6.7s
13s
©2015PearsonEducation,Inc.
QuickCheck 14.3
Atypicalearthquakeproducesverticaloscillationsoftheearth.Supposea
particularquakeoscillatesthegroundatafrequencyof0.15Hz.Astheearth
movesupanddown,whattimeelapsesbetweenthehighestpointofthe
motionandthelowestpoint?
◦
◦
◦
◦
1s
3.3s
6.7s
13s
©2015PearsonEducation,Inc.
OscillatoryMotion
Thegraphofanoscillatorymotionhastheformofacosinefunction.
Agraphorafunctionthathastheformofasineorcosinefunctioniscalledsinusoidal.
Asinusoidaloscillationiscalledsimpleharmonicmotion(SHM).
©2015PearsonEducation,Inc.
OscillatoryMotion
Text:p.440
©2015PearsonEducation,Inc.
Section14.2Linear
RestoringForcesand
SHM
©2015PearsonEducation,Inc.
LinearRestoringForcesandSHM
Ifwedisplaceagliderattachedtoa
springfromitsequilibriumposition,the
springexertsarestoringforceback
towardequilibrium.
©2015PearsonEducation,Inc.
LinearRestoringForcesandSHM
Thisisalinearrestoringforce;thenet
forceistowardtheequilibrium
positionandisproportionaltothe
distancefromequilibrium.
©2015PearsonEducation,Inc.
MotionofaMassonaSpring
Theamplitude A istheobject’s
maximumdisplacementfrom
equilibrium.
Oscillationaboutanequilibrium
positionwithalinearrestoring
forceisalwayssimpleharmonic
motion.
©2015PearsonEducation,Inc.
VerticalMassonaSpring
Forahangingweight,theequilibriumpositionoftheblockiswhereithangs
motionless.ThespringisstretchedbyΔL.
©2015PearsonEducation,Inc.
VerticalMassonaSpring
ThevalueofΔLisdeterminedbysolvingthestatic-equilibriumproblem.
Hooke’sLawsays
Newton’sfirstlawfortheblockinequilibriumis
Thereforethelengthofthespringattheequilibriumpositionis
©2015PearsonEducation,Inc.
VerticalMassonaSpring
Whentheblockisabovethe
equilibriumposition,thespring
isstillstretched byanamountΔL
– y.
Thenetforceontheblockis
• ButkΔL– mg=0,fromEquation14.4,sothenetforceon
theblockis
©2015PearsonEducation,Inc.
VerticalMassonaSpring
Theroleofgravityistodetermine
wheretheequilibriumpositionis,
butitdoesn’taffecttherestoring
forcefordisplacementfromthe
equilibriumposition.
Becauseithasalinearrestoring
force,amassonaverticalspring
oscillateswithsimpleharmonic
motion.
©2015PearsonEducation,Inc.
ThePendulum
Apendulum isamasssuspendedfroma
pivotpointbyalightstringorrod.
Themassmovesalongacirculararc.The
netforceisthetangentialcomponentof
theweight:
©2015PearsonEducation,Inc.
ThePendulum
Theequationissimplifiedforsmallangles
because
sinθ ≈ θ
Thisiscalledthesmall-angleapproximation.
Thereforetherestoringforceis
Theforceonapendulumisalinearrestoring
forceforsmallangles,sothependulumwill
undergosimpleharmonicmotion.
©2015PearsonEducation,Inc.
Section14.3Describing
Simple
HarmonicMotion
©2015PearsonEducation,Inc.
DescribingSimpleHarmonicMotion
1. Themassstartsatitsmaximum
positivedisplacement,y=A.The
velocityiszero,buttheaccelerationis
negativebecausethereisanet
downwardforce.
2. Themassisnowmovingdownward,so
thevelocityisnegative.Asthemass
nearsequilibrium,therestoringforce—
andthusthemagnitudeofthe
acceleration—decreases.
3. Atthistimethemassismoving
downwardwithitsmaximumspeed.
It’sattheequilibriumposition,sothe
netforce—andthustheacceleration—
iszero.
Text:p.443
©2015PearsonEducation,Inc.
DescribingSimpleHarmonicMotion
4. Thevelocityisstillnegativebut
itsmagnitudeisdecreasing,so
theaccelerationispositive.
5. Themasshasreachedthelowest
pointofitsmotion,aturning
point.Thespringisatits
maximumextension,sothereis
anetupwardforceandthe
accelerationispositive.
6. Themasshasbegunmoving
upward;thevelocityand
accelerationarepositive.
Text:p.443
©2015PearsonEducation,Inc.
DescribingSimpleHarmonicMotion
7. Themassispassingthroughthe
equilibriumpositionagain,inthe
oppositedirection,soithasa
positivevelocity.Thereisnonet
force,sotheaccelerationiszero.
8. Themasscontinuesmoving
upward.Thevelocityispositivebut
itsmagnitudeisdecreasing,sothe
accelerationisnegative.
9. Themassisnowbackatitsstarting
position.Thisisanotherturning
point.Themassisatrestbutwill
soonbeginmovingdownward,and
thecyclewillrepeat.
Text:p.443
©2015PearsonEducation,Inc.
DescribingSimpleHarmonicMotion
Theposition-versus-timegraphforoscillatorymotionisacosinecurve:
x(t)indicatesthatthepositionisafunction oftime.
Thecosinefunctioncanbewrittenintermsoffrequency:
©2015PearsonEducation,Inc.
DescribingSimpleHarmonicMotion
Thevelocitygraphisanupside-downsinefunctionwiththesameperiodT:
Therestoringforcecausesanacceleration:
Theacceleration-versus-timegraphisinvertedfromtheposition-versus-time
graphandcanalsobewritten
©2015PearsonEducation,Inc.
DescribingSimpleHarmonicMotion
Text:p.445
©2015PearsonEducation,Inc.
ConnectiontoUniformCircularMotion
Circularmotionandsimpleharmonic
motionaremotionsthatrepeat.
Uniformcircularmotionprojectedonto
onedimensionissimpleharmonic
motion.
©2015PearsonEducation,Inc.
ConnectiontoUniformCircularMotion
Thex-componentofthecircularmotionwhentheparticleisatangleϕ is x =Acosϕ.
Theangleatalatertimeisϕ= ωt.
ωistheparticle’sangularvelocity:ω=2πf.
©2015PearsonEducation,Inc.
ConnectiontoUniformCircularMotion
Thereforetheparticle’sx-componentisexpressed
x(t)=Acos(2pft)
Thisisthesameequationforthepositionofamassonaspring.
Thex-componentofaparticleinuniformcircularmotionissimple
harmonicmotion.
©2015PearsonEducation,Inc.
ConnectiontoUniformCircularMotion
Thex-componentofthevelocityvectoris
vx =-vsinϕ =-(2pf)Asin(2pft)
Thiscorrespondstosimpleharmonicmotionifwedefinethemaximumspeedtobe
vmax =2pfA
©2015PearsonEducation,Inc.
ConnectiontoUniformCircularMotion
Thex-componentoftheaccelerationvectoris
ax =-a cosϕ =-(2pf)2A cos(2pft)
Themaximumaccelerationisthus
amax =(2pf)2A
Forsimpleharmonicmotion,ifyouknowtheamplitudeandfrequency,
themotioniscompletelyspecified.
©2015PearsonEducation,Inc.
ConnectiontoUniformCircularMotion
Text:p.447
©2015PearsonEducation,Inc.
QuickCheck14.9
Amassoscillatesonahorizontalspring.It’svelocityisvx andthespringexertsforceFx.Atthe
timeindicatedbythearrow,
◦
◦
◦
◦
◦
vx is+ andFx is+
vx is+ andFx is–
vx is– andFx is0
vx is0andFx is+
vx is0andFx is–
©2015PearsonEducation,Inc.
QuickCheck14.9
Amassoscillatesonahorizontalspring.It’svelocityisvx andthespringexertsforceFx.Atthe
timeindicatedbythearrow,
◦
◦
◦
◦
◦
vx is+ andFx is+
vx is+ andFx is–
vx is– andFx is0
vx is0andFx is+
vx is0andFx is–
©2015PearsonEducation,Inc.
QuickCheck14.10
Amassoscillatesonahorizontalspring.It’svelocityisvx andthespringexertsforceFx.Atthe
timeindicatedbythearrow,
◦
◦
◦
◦
◦
vx is+ andFx is+
vx is+ andFx is–
vx is– andFx is0
vx is0andFx is+
vx is0andFx is–
©2015PearsonEducation,Inc.
QuickCheck14.10
Amassoscillatesonahorizontalspring.It’svelocityisvx andthespringexertsforceFx.Atthe
timeindicatedbythearrow,
◦
◦
◦
◦
◦
vx is+ andFx is+
vx is+ andFx is–
vx is– andFx is0
vx is0andFx is+
vx is0andFx is–
©2015PearsonEducation,Inc.
QuickCheck14.11
Ablockoscillatesonaverticalspring.Whentheblockisatthelowestpointoftheoscillation,it’s
accelerationay is
◦ Negative.
◦ Zero.
◦ Positive.
©2015PearsonEducation,Inc.
QuickCheck14.11
Ablockoscillatesonaverticalspring.Whentheblockisatthelowestpointoftheoscillation,it’s
accelerationay is
◦ Negative.
◦ Zero.
◦ Positive.
©2015PearsonEducation,Inc.
Example14.3Measuringtheswayofa
tallbuilding
TheJohnHancockCenterinChicagois100storieshigh.Strongwindscancausethebuildingto
sway,asisthecasewithalltallbuildings.Onparticularlywindydays,thetopofthebuilding
oscillateswithanamplitudeof40cm(≈16in)andaperiodof7.7s.Whatarethemaximum
speedandaccelerationofthetopofthebuilding?
©2015PearsonEducation,Inc.
Example14.3Measuringtheswayofa
tallbuilding
PREPARE Wewillassumethattheoscillationofthebuildingissimpleharmonicmotionwith
amplitudeA=0.40m.Thefrequencycanbecomputedfromtheperiod:
©2015PearsonEducation,Inc.
Example14.3Measuringtheswayofa
tallbuilding(cont.)
SOLVE WecanusetheequationsformaximumvelocityandaccelerationinSynthesis14.1to
compute:
vmax =2pfA =2p (0.13Hz)(0.40m)=0.33m/s
amax =(2pf)2A=[2p (0.13Hz)]2(0.40m)=0.27m/s2
Intermsofthefree-fallacceleration,themaximumaccelerationisamax =0.027g.
©2015PearsonEducation,Inc.
Example14.3Measuringtheswayofa
tallbuilding(cont.)
Theaccelerationisquitesmall,asyouwouldexpect;ifitwerelarge,buildingoccupants
wouldcertainlycomplain!Eveniftheydon’tnoticethemotiondirectly,officeworkersonhigh
floorsoftallbuildingsmayexperienceabitofnauseawhentheoscillationsarelargebecausethe
accelerationaffectstheequilibriumorganintheinnerear.
ASSESS
©2015PearsonEducation,Inc.
Section14.4Energyin
Simple
HarmonicMotion
©2015PearsonEducation,Inc.
EnergyinSimpleHarmonicMotion
Theinterplaybetweenkineticandpotentialenergyisveryimportantforunderstandingsimple
harmonicmotion.
©2015PearsonEducation,Inc.
EnergyinSimpleHarmonicMotion
Foramassonaspring,whentheobjectisat
restthepotentialenergyisamaximumand
thekineticenergyis0.
Attheequilibriumposition,thekinetic
energyisamaximumandthepotential
energyis0.
©2015PearsonEducation,Inc.
EnergyinSimpleHarmonicMotion
Thepotentialenergyforthemassonaspring
is
Theconservationofenergycanbewritten:
©2015PearsonEducation,Inc.
EnergyinSimpleHarmonicMotion
Atmaximumdisplacement,theenergyis
purelypotential:
Atx=0,theequilibriumposition,theenergyis
purelykinetic:
©2015PearsonEducation,Inc.
FindingtheFrequencyforSimple
HarmonicMotion
Becauseofconservationofenergy,themaximumpotentialenergymustbeequaltothe
maximumkineticenergy:
Solvingforthemaximumvelocitywefind
Earlierwefoundthat
©2015PearsonEducation,Inc.
QuickCheck14.7
Twoidenticalblocksoscillateondifferenthorizontalsprings.Whichspringhasthelargerspring
constant?
◦ Theredspring
◦ Thebluespring
◦ There’snotenough
informationtotell.
©2015PearsonEducation,Inc.
QuickCheck14.7
Twoidenticalblocksoscillateondifferenthorizontalsprings.Whichspringhasthelargerspring
constant?
◦ Theredspring
◦ Thebluespring
◦ There’snotenough
informationtotell.
©2015PearsonEducation,Inc.
QuickCheck14.8
Ablockofmassm oscillatesonahorizontalspringwithperiodT = 2.0s.Ifasecondidentical
blockisgluedtothetopofthefirstblock,thenewperiodwillbe
◦
◦
◦
◦
◦
1.0s
1.4s
2.0s
2.8s
4.0s
©2015PearsonEducation,Inc.
QuickCheck14.8
Ablockofmassm oscillatesonahorizontalspringwithperiodT = 2.0s.Ifasecondidentical
blockisgluedtothetopofthefirstblock,thenewperiodwillbe
◦
◦
◦
◦
◦
1.0s
1.4s
2.0s
2.8s
4.0s
©2015PearsonEducation,Inc.
FindingtheFrequencyforSimple
HarmonicMotion
Thefrequencyandperiodofsimple
harmonicmotionaredeterminedbythe
physicalpropertiesoftheoscillator.
Thefrequencyandperiodofsimple
harmonicmotiondonotdependonthe
amplitudeA.
©2015PearsonEducation,Inc.
QuickCheck14.4
Ablockoscillatesonaverylonghorizontalspring.Thegraphshowstheblock’skineticenergyas
afunctionofposition.Whatisthespringconstant?
◦
◦
◦
◦
1N/m
2N/m
4N/m
8N/m
©2015PearsonEducation,Inc.
QuickCheck14.4
Ablockoscillatesonaverylonghorizontalspring.Thegraphshowstheblock’skineticenergyas
afunctionofposition.Whatisthespringconstant?
◦
◦
◦
◦
1N/m
2N/m
4N/m
8N/m
©2015PearsonEducation,Inc.
QuickCheck14.5
Amassoscillatesonahorizontalspringwithperiod
T = 2.0s.Iftheamplitudeoftheoscillationisdoubled,
thenewperiodwillbe
◦
◦
◦
◦
◦
1.0s
1.4s
2.0s
2.8s
4.0s
©2015PearsonEducation,Inc.
QuickCheck14.5
Amassoscillatesonahorizontalspringwithperiod
T = 2.0s.Iftheamplitudeoftheoscillationisdoubled,
thenewperiodwillbe
◦
◦
◦
◦
◦
1.0s
1.4s
2.0s
2.8s
4.0s
©2015PearsonEducation,Inc.
Section14.5Pendulum
Motion
©2015PearsonEducation,Inc.
PendulumMotion
Thetangentialrestoringforcefora
pendulumoflengthL displacedby
arclengths is
Thisisthesamelinearrestoring
forceasthespringbutwiththe
constantsmg/L insteadofk.
©2015PearsonEducation,Inc.
PendulumMotion
Theoscillationofapendulum
issimpleharmonicmotion;
theequationsofmotioncan
bewrittenforthearclength
ortheangle:
s(t)=A cos(2πft)
or
θ(t)=θmax cos(2πft)
©2015PearsonEducation,Inc.
PendulumMotion
Thefrequencycanbeobtained
fromtheequationforthe
frequencyofthemassonaspring
bysubstitutingmg/L inplace
ofk:
©2015PearsonEducation,Inc.
PendulumMotion
Asforamassonaspring,the
frequencydoesnotdependonthe
amplitude.Notealsothatthe
frequency,andhencetheperiod,is
independentofthemass. Itdepends
onlyonthelengthofthependulum.
©2015PearsonEducation,Inc.
QuickCheck14.15
Apendulumispulledto
thesideandreleased.
Themassswingstothe
rightasshown.The
diagramshowspositionsforhalfofacompleteoscillation.
1.Atwhichpointorpointsisthespeedthehighest?
2.Atwhichpointorpointsistheaccelerationthe
greatest?
3.Atwhichpointorpointsistherestoringforcethegreatest?
©2015PearsonEducation,Inc.
QuickCheck14.15
Apendulumispulledto
thesideandreleased.
Themassswingstothe
rightasshown.The
diagramshowspositionsforhalfofacompleteoscillation.
C
1.Atwhichpointorpointsisthespeedthehighest?
2.Atwhichpointorpointsistheaccelerationthe
A,E
greatest?
3.Atwhichpointorpointsistherestoringforcethegreatest?
A,E
©2015PearsonEducation,Inc.
QuickCheck 14.16
Amassontheend
ofastringispulled
tothesideandreleased.
1.Atwhichtimeortimesshownistheacceleration
zero?
2.Atwhichtimeortimesshownisthekineticenergy
amaximum?
3.Atwhichtimeortimesshownisthepotentialenergy
amaximum?
©2015PearsonEducation,Inc.
QuickCheck 14.16
Amassontheend
ofastringispulled
tothesideandreleased.
1.Atwhichtimeortimesshownistheacceleration
zero?
2.Atwhichtimeortimesshownisthekineticenergy
amaximum?
3.Atwhichtimeortimesshownisthepotentialenergy
amaximum?
©2015PearsonEducation,Inc.
B,D
B,D
A,C,E
QuickCheck14.17
Aballonamassless,rigidrodoscillatesasasimplependulumwithaperiodof2.0s.Iftheballis
replacedwithanotherballhavingtwicethemass,theperiodwillbe
◦
◦
◦
◦
◦
1.0s
1.4s
2.0s
2.8s
4.0s
©2015PearsonEducation,Inc.
QuickCheck14.17
Aballonamassless,rigidrodoscillatesasasimplependulumwithaperiodof2.0s.Iftheballis
replacedwithanotherballhavingtwicethemass,theperiodwillbe
◦
◦
◦
◦
◦
1.0s
1.4s
2.0s
2.8s
4.0s
©2015PearsonEducation,Inc.
QuickCheck14.18
OnPlanetX,aballonamassless,rigidrodoscillatesasasimplependulumwithaperiodof2.0s.
Ifthependulumis
takentothemoonofPlanetX,wherethefree-fallaccelerationg ishalfas
big,theperiodwillbe
◦
◦
◦
◦
◦
1.0s
1.4s
2.0s
2.8s
4.0s
©2015PearsonEducation,Inc.
QuickCheck14.18
OnPlanetX,aballonamassless,rigidrodoscillatesasasimplependulumwithaperiodof2.0s.
Ifthependulumis
takentothemoonofPlanetX,wherethefree-fallaccelerationg ishalfas
big,theperiodwillbe
◦
◦
◦
◦
◦
1.0s
1.4s
2.0s
2.8s
4.0s
©2015PearsonEducation,Inc.
QuickCheck14.19
Aseriesofpendulumswithdifferentlengthstringsanddifferentmassesisshown
below.Eachpendulumispulledtothesidebythesame(small)angle,thependulums
arereleased,andtheybegintoswingfromsidetoside.
Whichofthependulumsoscillateswiththehighestfrequency?
©2015PearsonEducation,Inc.
QuickCheck14.19
Aseriesofpendulumswithdifferentlengthstringsanddifferentmassesisshown
below.Eachpendulumispulledtothesidebythesame(small)angle,thependulums
arereleased,andtheybegintoswingfromsidetoside.
A
Whichofthependulumsoscillateswiththehighestfrequency?
©2015PearsonEducation,Inc.