oscillator 53.7 hmmm frictionlessm307/heather...harmonic oscillator 53.7 hmmm f frictionless 7 to u...
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Harmonic Oscillator 53.7
HMMM f frictionless7
to u displacement fromequilibrium equilibriumposition
positive when stretched
11M U Fspring KU Hooke's law
Is kg s2
Tmu t k U Ou Im U Ou t wo2u o Wo Fml
charegg r't wi or2 Wo2r I Iwo
U C cos wot t Czsin wot
Experiment 1 to find k
Take m 1kg pull 1 unit to the rightrelease from rest
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U o I U lo O
sdu f u cos fit
Then record how long it takes to arrive at
equilibrium for the first time Say it tookt 0.5 Sec
O U 0.5 co's TKO 5
TK0.5 721
ik TLk Tv kg s2
Experiment2forfindinglet
IFspring KL
Mm
at rest vftp.auity m8
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KL mgkmTf End of lecture 1
Hanging Harmonic Oscillator
y om
f y L Mm u o downward stretchedu y L is positive
at rest y ing
ult o is a constant equilibrium soln
U displacement from equilibrium
fthis
KU
mu t Ku O
U C U t Cz U2
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mdaff Fgravity t Fspring Mg Ky
my t Ky my
mg y oCompare y and U2
u y mfs so y u 1M
m utmost klutz mgmu t ka t my my
mu ku 0
Notice Mg Ky o yA Mfd is an
equilibrium so In
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Harmonicoscillatormu t k U
Wo FknU C cos wot t Czsin wot
Acrosscostwot Asin8 sin l wotC Cz
i
Ult A cos wot d
CartesignoryAd polar coord
IZ C Cz Acos8 AsiaAsinof www.ag
A 1determined by the initial
t socondition
standard arctan C f It ILI2
mined Amplitude A It fA TAIIIalpha tant Econditions
qThthis carries less infothan
these so need to also knowsigns of Ci
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tano
areIi ii
i orI
Ito a arctan Eo d arctan ft TL
If C1 0 d It if Cz OTiz if CzC 0
Ext ult 4 cosGat t since t
A FEE Etf
4 ItNz
4 341aretan ta aretanf 1 t TLc
UH cos Gat 3
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Graph ult A cos wot 8
ACOS X
1
VIIcos x Cos x 12A
ult A cos wot 8
k F woA
Acost
To
iI
t
A o
velocity o when u Acomment added after the lecture
this is assuming we choseFirst peak happens when Wot 8 0 2 8 2x so the smallesti e t I t o sit Wot 5 2 InWo happens when wot 8 0
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Period Cos wot 8 cos wot 8 2E
cos two ft 1 8
T 2 2E FFNatural Wo fknFrequency
Extcontide Graph ult cos Gat 3AUCt
t
I
First peak when sat 3 O2kt 3
t IgPeriod T 2 3 I
f
There are many examples of vibration in natureEnd of lecture 2
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Pendulummd4 masinodt2ii Ld20
It gsinoi
ii To O EsinO nonlinear
kmgsi.noFgrauI m'S
small oscillations sin 0 0
O tIO Linearization
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