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SimpleHarmonicMotionAmassonaspringisanexample
ofperiodicmotion.
Intheabsenceoffrictionand
dragforces,itoscillatesbetween
itsreleasepointandapoint
equa y ar romequ r um.
Thisiscalledsimpleharmonic
motion. Therestoringforcehas
thesimplestpossibleform:
HOOKESLAW
F= kxnote: minussignjusttellsusthatthe
forceopposesthedisplacement
BestIgnored
Useyourphysicsintuition!
Forceconstantk Tellsushowstiffthespringis.
Stiffspring largek
Units:N/m [NewtonperMeter]
Example:
A0.5102kg masshangs
fromaspringandis
foundtostretchthe
unstretched
length
springby0.25m fromits
unstretchedlength.
Whatisthespring
constantofthespring?
equilibrium
stretch(d)
equilibrium
position
SimpleHarmonicMotionDisplacement(x) isthedirected
distanceoftheobjectfrom
equilibrium.
Amplitude(A) isthemaximum
displacement.
Period(T) isthetimeforonefull
cycle.
Frequency(f) isthenumberoffull
cyclespersecond.
Thinkaboutitthisway:
f=1/T T=1/for
Ablockonaspringbouncesupanddown
every0.200seconds. Howmanytimesdoes
itbounceupanddowninonesecond?
T=0.200s
f=1/T=5timespersecond
f=5Hz(hertz)
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Bigpoint:
TheperiodT (orfrequencyf)ofanoscillatoris
relatedonlytothestiffness k oftherestoring
forceandtotheinertiam ofthesystem.
Itdoesnottellyouanythingabouthowthe
systemgotstartedandisindependentofthe
amplitudeA.
Howdoweknowthis? Letscheckitout!
kxdt
xdmmaF
2
2
===N2:
m
kwith)tcos(A)t(x =+=
Assumesolution:
0xm
k
dt
xd2
2
=+
)tsin(Adt
)t(dx+=
)t(xm
k)t(x)tcos(A
dt
)t(xd 222
2
==+=
kxdt
xdmmaF
2
2
===
m
kwith)tcos(A)t(x =+=
N2:
Assumesolution:
0xm
k
dt
xd2
2
=+
oscillatorharmonic0)t(xm
k
dt
)t(xd2
2
=+
k
m2Tand
m
k
2
1ff2
m
k=
===
Whatwefind:
kmT 2=Period:
mkTf
2
1/1 ==
In epen ento ampl tu e
(naturalfrequency)Frequency:
Problem:
A2.00kg blockhangsfromaspringandoscillateswith
aperiod T=1.00s. Whatmassofblock,hungfromthe
samespring,wouldhaveaperiodof T=2.00s?
mT 2
k=
.
EnergyofSimpleHarmonicMotionPotentialenergyofaspring:
Elasticforcesareconservative,
therefore,thetotalmechanicalenergyisconserved.
Itiseasiesttocalculatethetotalenergyatthe
endpointsofmotion,wherethekineticenergyis
zeroandtheenergyisallpotentialenergy.
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SimpleHarmonicMotionThetotalenergyofanobjectinsimpleharmonicmotionisdirectlyproportionaltothesquareoftheamplitudeoftheobjectsdisplacement.
SimpleHarmonicMotionThisallowsustocalculatethevelocityasafunction
ofposition:
kA2 =kx2 +mv2 .....
andthemaximumvelocity(atx=0):
SimpleHarmonicMotionTheenergyvariesfrombeingcompletelykineticto
completelypotential,andbackagain.
EquationsofMotionAnequationofmotiongivesthepositionofanobject
asafunctionoftime.
Simpleharmonicmotioncanberepresentedasa
componentofuniformcircularmotion:(xldemo)
EquationsofMotionTheequationofmotionfortheoscillatingobject
couldbe givenby:
ere, eon yposs y s a y= a = . ore
likely,wewouldwanty=A att=0;thatis,att=0the
objectwouldhaveitsmaximumdisplacement.Inthat
case,
Puzzle! A mass attached to a spring oscillates back and
forth as indicated in the position vs. time plot below. At
point
P, the mass has:
P
t
x
2
x A cos( t)
v A sin( t)
a A cos( t)
=
=
=
1. positive velocity and positive acceleration.
2. positive velocity and negative acceleration.
3. positive velocity and zero acceleration.
4. negative velocity and positive acceleration.
5. negative velocity and negative acceleration.
6. negative velocity and zero acceleration.
7. zero velocity but is accelerating (positively or
negatively).
8. zero velocity and zero acceleration.
