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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+,2 Si"ple Har"onic otion
' "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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+,2 Si"ple Har"onic otion
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
otion and Si"ple Har"onic otion
'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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+,, *onnections between Unifor" *ircular
otion and Si"ple Har"onic otion
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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+,, *onnections between Unifor" *ircular
otion and Si"ple Har"onic otion
The position as a function of ti"e3
The angular fre/uency3
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+,, *onnections between Unifor" *ircular
otion and Si"ple Har"onic otion
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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+,< .nergy *onservation in )scillatory
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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+,< .nergy *onservation in )scillatory
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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+,7 !a"ped )scillations
This e&ponential decrease is shown in the
figure3
+, 7 ! d ) ill ti
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+,7 !a"ped )scillations
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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Su""ary of *hapter +,
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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Su""ary of *hapter +,
'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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Su""ary of *hapter +,
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
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Su""ary of *hapter +,
Period of a si"ple pendulu"3
Period of a physical pendulu"3
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Su""ary of *hapter +,
)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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Su""ary of *hapter +,
'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"
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