PHYS207-OscillationsRecitation-J.Hedberg-2016
Recitation5,Phys207
1.Oscillations
1.1Oscillationsareeverywhere:
Comeupwith5examplesofoscillatingphenomena.Estimatetheperiod[inwhateverunitseemsmostappropriate].(For
instance:mybeatingheartcontractsandexpandsperiodically,andhasaperiodofapproximately1second)
1.2Oscillationgraphs
Belowareplotsofthepositionofanobjectwithrespecttotime.Foreachplot:constructacorrespondingv(t)(velocityasa
functionoftimegraph)forthismotion.Also,answerthefollowingquestionsregardingthismotion:
1. Isitperiodic?
2. Isitsimpleharmonicmotion?
3. Whatistheperiodofthismotion?
4. Whatisthefrequencyofthemotion?
x [m]
x [m]
2
2
1
1
0
0
2
4
6
8
2
t [s]
-1
-1
-2
-2
4
6
8
t [s]
x [m]
2
1
0
2
4
6
8
t [s]
-1
-2
1.3Simpleharmonicmotion
Hereisthepositionw.r.ttimegraphforanobjectundergoingSimpleHarmonicMotion.(positiveXistotherightoftheorigin)
x [m]
2
1
0
2
4
6
8
t [s]
-1
-2
a)Atwhatpoint(s)intimeistheobjectstationary?
b)Wherealongthex-axisdoestheobjectcometorest?
c)Atwhatpoint(s)intimeistheobjectmovingtotherightwithitsmaximumspeed?
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PHYS207-OscillationsRecitation-J.Hedberg-2016
d)Atwhatpoint(s)intimeistheobjectmovingtotheleftwithitsmaximumspeed?
e)Atwhatpoint(s)intimedoestheobjecthavethelargest,positive,acceleration?
f)Graphicallyestimatethemaximumvelocityofthisobject.(whatistheslopewhenthelinecrossesx=0)
g)Calculatetheactualvmax byfindingtheperiodandamplitudefromthegraphs.
1.4YoudrawSHM
Drawquantitativeplotsofx(t),v(t)anda(t)foranobjectundergoingSHM.
a)Itsfrequencyis4Hzandamplitudeofmotionis2meters.Theobjecthasitslargestpositivevelocitywhenitpassesthrough
theoriginatt=0.
b)Itsperiodis10sandamplitudeofmotionis30meters.Theobjecthasitslargestnegativeaccelerationwhenitpassesthrough
theoriginatt=0.
1.5Spring-Mass
Hereisamassattachedtoaspringsittingontheground.Theinterfacebetweenthemassandthegroundisfrictionless.Att=0,
themassisreleasedfromadistancedtotheright(+)oftheequilibriumposition.ThemotionthatresultsisSHMandhasa
periodofT0 seconds.
a)Drawafreebodydiagramofthemassatt = 0.
b)Drawthefreebodydiagramofthemassatt=T/4,T/2,3T/4,andT .
c)Atthesetimes,indicategraphicallythedirectionsofthevelocityandaccelerationvectorsforthemass.
d)Writeequationsthatdescribetheposition,velocityandaccelerationofthemassasfunctionsoftime.
e)Makeplotsofthesethreeequations.
1.6SHMinequations
Thefollowingequationdescribesanobject'spositionwhileundergoingsimpleharmonicmotion.
x(t) = A cos(
2π
t)
T
a)WhatdoestheAinthisequationrepresent?Whatareitsdimensions?
b)Whatdoestheargumentofthecosinefunction( 2π
t)represent?Whatdimensions/unitsisit?
T
c)When(i.e.atwhatt)willthisfunctionproduceitsmaximum(mostpositive)values?It'sminimum(mostnegative)values?
Whenwillitequalzero?
d)Findanequationthatusesthesin()functioninsteadthatwillgeneratetheexactsamepositionsasafunctionoftime.(Usea
phasefactor)
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PHYS207-OscillationsRecitation-J.Hedberg-2016
1.7Allthedetailsofanoscillator
A400grammassisattachedtoaspring.Itoscillateswithafrequencyof3Hz.Att=0,themassislocated0.01mawayfromthe
equilibriumandhasavelocityof-0.3m/s.Determine:
a)Theperiod
b)Theangularfrequency
c)Theamplitudeofmotion
d)Thephaseconstant
e)Themaximumspeed
f)Thetotalenergyofthesystem
g)Themaximumacceleration
h)Thepositionatt = 2seconds.
1.8Readingagraph
Basedonthegraphbelowthatshowsboththepotentialenergyandtotalmechanicalenergyasfunctionsofdistanceforthe
spring-masssystem:
1. Whatistheequilibriumlengthofthespring?
2. Wherewillthevelocityofthemassbezero?
3. Whatisthemaximumkineticenergythatthismasswillhave?
4. Whatisthespringconstantofthespringinthissystem?
Energy [J]
Potential Energy
30
20
Total Energy
10
0
2
6
10
14
18 x [cm]
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