Download - Chem 373- Harmonic oscillator...classical
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Harmonic oscillator...classical
Let us consider a particle of mass m attached to a spring
Equilibrium
x=0,t=0
o
x
Stretchx=xo
compressx=-xoxo
xo
At the beginning at t = o the particle is at equilibrium,
that is no particle is working at it , F = 0,
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Harmonic oscillator...classical
In general F = -k x . The force propotional to
displacement and pointing in opposite directiono
k is the force constant of the spring
Equilibrium
x=0,F=0
o
x
xo
xo
xo F=-kxo
xo
F= kxo
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We might consider as an other example two particles
attached to each side of a spring
re
A B
F= 0 Equilibriumr = re
Case I: Equilibrium
A B
F= -kx Stretchr = re+x
Case II: Stretch
r = re+x
Harmonic oscillator...classical
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Harmonic oscillator...classical
re-x
A B
F= -k(-x) Equilibrium
r = re
Case III: Compress
x
Again we have that the force F is proportionalto the displacement x
and pointing in the opposite direction
F = - k x
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Harmonic oscillator...classical
Let us look at this solution
(a) for or t = o we have x
for or t =2
x
o
o
k
mt
b km
t mk
A
= =
=
=
0 0
2
( )
( )
( )
( )
c km
t mk
o
dk
m tm
k A
e
k
m t
m
k o
for or t = x
for3
or t =3
x
for or t = 2 x
o
o
o
=
=
=
=
=
=
2 2
2
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Equilibrium
x=0,t=0
o
x
Stretch
x=xo
compress
x=-xoxo
xo
position x = A sin (k
mt )
Harmonic oscillator...classical
A
x
t
2
m
k
m
k
3
2
m
k
2 mk
-A
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Ak
m
v
t
2
mk
m
k
3
2
m
k
2 mk
-Am
k
velocity v =dx
dt= A
k
mcos (
k
mt )
Harmonic oscillator...classical
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Ak
m
v
t
2
m
k
m
k
3
2
m
k
2m
k
-Am
k
Force = - k x = -Ak
msin (
k
mt )
Harmonic oscillator...classical
Equilibrium
x=0,F=0
o
x
xo
xo
xo F=-kxo
xo
F= kxo
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Harmonic oscillator...classical
It
m
k
as
k
m
follows that the time to complete one cycle is
t
a consequence one can complete
=t
cycles per time unit
cycle
cycle
=
=
2
1 1
2
The frequency is often written
as =2
is referred to as the circular frequency.
We clearly have
=
k
m
where
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Harmonic oscillator...classical
We might also look at the kinetic energy
T =1
2t
t
t
mv m Ak
m
k
m
T mA km
km
T A kk
m
22
2 2
2 2
1
2
12
1
2
=
=
=
cos
cos
cos
t
2
m
k
m
k
3
2
m
k
2m
k1
2A2k
T
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Harmonic oscillator...classicalWhat about potential energy
V(x) ??
We always have
F = -dV( x)
d x
F = -k x = -dV( x)
d x
dV
d xd x = V( x) - V(o)
= x d x = 12
x
V( x) = 12
x
x
x 2
2
Thus
or
k k o
Thus
k
0
0
V(x) = 1/2k2x2V
x-A2 A2k1 > k2
E
V(x) = 1/2k1x2
-A1 A1
t
2
m
k
mk
3
2
m
k
2m
k1
2 A2
k
V
V =1
2A2k sin2
k
mt
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Harmonic oscillator...classical
E T V A k km
A k km
E A k
We
= + =
+
=
12
12
1
2
2 2 2 2
2
cos cost t
note total energy independent of t
V(
x) = 1/2k
2x2
V
x-A2 A2k1 > k2
E
V(x) = 1/2k1
x2
-A1 A1
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Harmonic oscillator...classicalFrom
k
m m
Thus
Note
depends
the relation
=
1
2 k = 4
E = 2 m A
that the amplitude A
A =2E
k
on E and k. For agiven E the smaller k the
larger A.
Note that the frequency isindependent of A
2
2 2
V(x) = 1/2k2x2V
x-A2 A2k1 > k2
E
V(x) = 1/2k1x2
-A1 A1