walker3 lecture ch13

8/10/2019 Walker3 Lecture Ch13

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2007 Pearson Prentice Hall

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(ecture )utlines

*hapter +,

Physics, 3rdEdition

-a"es S $alker


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*hapter +,

)scillations about

./uilibriu"


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Units of *hapter +,

Periodic otion

Si"ple Har"onic otion

*onnections between Unifor" *ircular

otion and Si"ple Har"onic otion

The Period of a ass on a Spring

.nergy *onservation in )scillatory

otion


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Units of *hapter +,

The Pendulu"

!a"ped )scillations

!riven )scillations and 1esonance


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+,+ Periodic otion

Period3 ti"e re/uired for one cycle of periodic"otion

4re/uency3 nu"ber of oscillations per unit ti"e

This unit is

called the Hert53


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+,2 Si"ple Har"onic otion

' spring e&erts a restoring force that is

proportional to the displace"ent fro"

e/uilibriu"3


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' "ass on a spring has a displace"ent as a

function of ti"e that is a sine or cosine curve3Here6Ais called

the a"plitude of

the "otion


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f we call the period of the "otion T8 this is the

ti"e to co"plete one full cycle 8 we can writethe position as a function of ti"e3

t is then straightforward to show that the

position at ti"e t + Tis the sa"e as theposition at ti"e t6 as we would e&pect


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+,, *onnections between Unifor" *ircular


'n ob9ect in si"ple

har"onic "otion has the

sa"e "otion as one

co"ponent of an ob9ectin unifor" circular

"otion3


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Here6 the ob9ect in circular "otion has an

angular speed of

where Tis the period of "otion of the

ob9ect in si"ple har"onic "otion


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The position as a function of ti"e3

The angular fre/uency3


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The velocity as a function of ti"e3

'nd the acceleration3

:oth of these are found by takingco"ponents of the circular "otion /uantities


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+,; The Period of a ass on a Spring

Since the force on a "ass on a spring is

proportional to the displace"ent6 and also tothe acceleration6 we find that

Substituting the ti"e dependencies of aandx

gives


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+,; The Period of a ass on a Spring

Therefore6 the period is


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+,< .nergy *onservation in )scillatory

otion

n an ideal syste" with no nonconservative

forces6 the total "echanical energy is

conserved 4or a "ass on a spring3

Since we know the position and velocity as

functions of ti"e6 we can find the "a&i"u"

kinetic and potential energies3


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otion

's a function of ti"e6

So the total energy is constant= as the

kinetic energy increases6 the potential

energy decreases6 and vice versa


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otion

This diagra" shows how the energy

transfor"s fro" potential to kinetic and

back6 while the total energy re"ains the

sa"e


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+,> The Pendulu"

' si"ple pendulu" consists of a "ass m#of

negligible si5e% suspended by a string or rod of

lengthL#and negligible "ass%

The angle it "akes with the vertical varies with

ti"e as a sine or cosine


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+,> The Pendulu"

(ooking at the forces

on the pendulu" bob6

we see that the

restoring force is

proportional to sin 6whereas the restoring

force for a spring is

proportional to the

displace"ent #whichis in this case%


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+,> The Pendulu"

However6 for s"all angles6 sin and are

appro&i"ately e/ual


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+,> The Pendulu"

Substituting for sin allows us to treat the

pendulu" in a "athe"atically identical way tothe "ass on a spring Therefore6 we find that the

period of a pendulu" depends only on the

length of the string3

+, > Th P d l


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+,> The Pendulu"

' physical pendulu" is a

solid "ass that oscillates

around its center of "ass6

but cannot be "odeled as apoint "ass suspended by a

"assless string .&a"ples3

+, > Th P d l


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+,> The Pendulu"

n this case6 it can be shown that the period

depends on the "o"ent of inertia3

Substituting the "o"ent of inertia of a point

"ass a distance lfro" the a&is of rotation

gives6 as e&pected6

+, 7 ! d ) ill ti


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+,7 !a"ped )scillations

n "ost physical situations6 there is a

nonconservative force of so"e sort6 which willtend to decrease the a"plitude of the

oscillation6 and which is typically proportional

to the speed3

This causes the a"plitude to decrease

e&ponentially with ti"e3

+, 7 ! d ) ill ti


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This e&ponential decrease is shown in the

figure3

+, 7 ! d ) ill ti


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The previous i"age shows a syste" that is

underda"ped 8 it goes through "ultipleoscillations before co"ing to rest ' critically

da"ped syste" is one that rela&es back to the

e/uilibriu" position without oscillating and in

"ini"u" ti"e= an overda"ped syste" will

also not oscillate but is da"ped so heavily

that it takes longer to reach e/uilibriu"

+, ? ! i ) ill ti d 1


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+,? !riven )scillations and 1esonance

'n oscillation can be driven by an oscillating

driving force= the fre/uency of the driving force

"ay or "ay not be the sa"e as the naturalfre/uency of the syste"


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+,? !riven )scillations and 1esonance

f the driving fre/uency

is close to the natural

fre/uency6 the

a"plitude can beco"e/uite large6 especially

if the da"ping is s"all

This is called

resonance


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Su""ary of *hapter +,

Period3 ti"e re/uired for a "otion to go

through a co"plete cycle

4re/uency3 nu"ber of oscillations per unit ti"e

'ngular fre/uency3

Si"ple har"onic "otion occurs when the

restoring force is proportional to the

displace"ent fro" e/uilibriu"


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The a"plitude is the "a&i"u" displace"ent

fro" e/uilibriu"

Position as a function of ti"e3

@elocity as a function of ti"e3


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'cceleration as a function of ti"e3

Period of a "ass on a spring3

Total energy in si"ple har"onic "otion3


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Potential energy as a function of ti"e3

Ainetic energy as a function of ti"e3

' si"ple pendulu" with s"all a"plitudee&hibits si"ple har"onic "otion


33/35


Period of a si"ple pendulu"3

Period of a physical pendulu"3


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)scillations where there is a nonconservative

force are called da"ped

Underda"ped3 the a"plitude decreases

e&ponentially with ti"e3

*ritically da"ped3 no oscillations= syste"

rela&es back to e/uilibriu" in "ini"u" ti"e

)verda"ped3 also no oscillations6 but

slower than critical da"ping


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'n oscillating syste" "ay be driven by an

e&ternal force

This force "ay replace energy lost to friction6

or "ay cause the a"plitude to increase greatly

at resonance

1esonance occurs when the driving fre/uency

is e/ual to the natural fre/uency of the syste"

walker3 lecture ch13

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