Review_Exam2

Review_Exam2 - Keys
 •  Thornton
&
Marion
and
Boas.
 •  Homework
problems
(solu<ons
at
culearn.
 • 

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Unformatted text preview: Keys
 •  Thornton
&
Marion
and
Boas.
 •  Homework
problems
(solu<ons
at
culearn).
 •  Clicker
ques<ons
(at
culearn
as
well).
 1)
Complex
number
and
variable
 •  z=x+iy,
or
rcosθ+isinθ,
or
reiθ •  Complex
conjugate,
z=x‐iy
is
conjugate
to
z=x +iy.
 •  Define
sinz
and
cosz.
 •  Roots
of
z3=8.
 Solu<on
of
2nd
linear
ODE
 •  Homogeneous
equa<on.
x”+2bx’+cx=0.
 




x(t)=Ae[‐b+sqrt(b^2‐ac)]t+Be[‐b‐sqrt(b^2‐ac)]t.
 




For
b^2‐ac<0,
oscilla<on.

 




x(t)=Ae[‐b+isqrt(ac‐b^2)]t+Be[‐b‐isqrt(ac‐b^2)]t



or


 




x(t)=Ae‐btcos(ωt‐δ).
 •  Inhomogeneous
equa<on.
x”+2bx’+cx=f(t)
 




x(t)=xc(t)+xp(t)
 



If
f(t)=dent,
xp(t)
can
be
constructed,
depending
 on
the
rela<on
of
n
to
the
roots
of
characteris<c
 equa<ons.


 Spring‐block
Oscillator
(linear,
for
small
 amplitude
oscilla<ons)
 •  F=mx”
or
–kx=mx”
or
x”+(k/m)x=0.
 Angular
frequency
ω0=sqrt(k/m),
period
T,
…
 x(t)=Asin(ω0t‐δ).





v=x’=Aω0cos(ω0t‐δ).
 E=T+U,
conserved.
 •  Phase
diagram
x
versus
v
or
x’.
 




An
ellipse
for
a
given
E
(no
damping).
 •  Electric
circuits,
a
pendulum,
and
a
floa<ng
 block,
….
 Spring‐block
(linear)
Oscillator
with
 damping
 •  
–kx‐bx’=mx”
or
x”+(b/m)x’+(k/m)x=0.
 



1)
Overdamping,
2)
cri<cal
damping,
and

 



3)
underdamping.
 



For
underdamping,
x(t)=Ae‐βtcos(ω1t‐δ).






 



E=T+U,
not
conserved.
dE/dt<0.
 •  Phase
diagram
x
versus
v
or
x’.
 Goes
to
the
center
as
t‐>
infinite.

 Spring‐block
(linear)
Oscillator
with
 damping
and
driving
force
 •  x”+(b/m)x’+(k/m)x=f(t).

[e.g.,
f(t)=Fcos(ωt)]
 



x(t)=xc(t)+xp(t)
 






[e.g.,
xp(t)=Dcos(ωt‐δ)]






 



Resonance:
when
ω
=ωR,
D
is
maximized.
 •  Define
Q=ωR/(2β)
–
quality
factor.
 Fourier
analysis
 •  Periodic
func<on,
f(t),
can
be
expressed
as
 sum
of
cosine
and
sine
func<ons
of
different
 periods.
 

f(t)=a0/2+a1cos(2πt/T)+a2cos(4πt/T)
+…
 
















+b1sin(2πt/T)+b2sin(4πt/T)
+…
 Green’s
func<on
method
 •  x(t)=G(t,t ’)
is
the
oscillator’s
response
at
<me
t
 to
an
impulse
force
at
<me
t ’.
Then
the
 response
of
the
oscillator
to
an
arbitrary
force
 f(t)
is

 




x(t)
=
integrate
f(t’)G(t,t’)dt ’
for
t ’
from
0
to
t.
 Chaos
 •  Nonlinear
oscilla<on
(olen
occurs
when
 oscilla<on
has
a
large
amplitude).

 •  Phase
diagrams
for
nonlinear
oscilla<on.
 ...
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This note was uploaded on 11/10/2009 for the course PHYS 2210 at Colorado.

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