12
11.17
k
k
2m
m
x
2
x
1
Note: As discussed in Section 3.2, the effect of any constant external
force on a harmonic oscillator is to shift the equilibrium position.
x
1
and
x
2
are the positions of the harmonic oscillator masses away from
their respective “shifted” equilibrium positions.
()
22
12
11
2
Tm
x
m
x
=+
±±
2
2
Vk
x k
xx
=+ −
1
LTV
=−
2
,
dL
L
mx
kx
k x
x
dt
x
x
∂∂
==
−
+
±
1
−
2
20
mx
kx
kx
+−=
2,
L
mx
k x
x
dt
x
x
−
±
1
−
221
mx
kx
kx
The secular equation (11.4.12) is thus
2
2
2
0
2
mk
k
km
k
ω
−+
−
=
−−
+
24
2
2
2
25 2
mm
kk
k
ωω
+
0
=
The eigenfrequencies are thus …
2
51
7
4
k
m
±
=
The homogeneous equations (Equations 11.4.10) for the two components of the j
th
eigenvector are …
2
1
2
2
2
0
2
j
j
a
k
a
k
−
=
+
For the first eigenvector (the antisymmetric mode, j = 1) …
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This note was uploaded on 01/06/2012 for the course PHYSICS 4360C taught by Professor Johnanderson during the Fall '11 term at UNF.
 Fall '11
 JohnAnderson
 Force, Mass, Work

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