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

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

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

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

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

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

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

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

Pendulummd4 masinodt2ii Ld20

It gsinoi

ii To O EsinO nonlinear

kmgsi.noFgrauI m'S

small oscillations sin 0 0

O tIO Linearization