Physics 325
Lecture 11
Oscillations
For the next few lectures we will discuss oscillations in some detail.
We have already
touched on the importance of simple harmonic motion when discussing the Taylor
expansion of a potential about its minimum
x
0
,
0
0
2
2
0
0
0
2
1
2
x
x
dU
d U
U
x
U
x
x
x
x
x
dx
dx
We are free to define
U
(
x
0
)
0
, and the first derivative vanishes by the condition that
x
0
is a minimum of the potential. The first non-zero term then is the second derivative which
is a Hooke’s law spring potential (letting
x
0
0
):
0
2
2
2
1
,
where
2
x
d U
U
x
kx
k
dx
Recall that
k
is positive in this case since
x
0
is a minimum. Simple harmonic motion is
extremely important because many potential functions can be successfully Taylor
expanded about their minimum, and the first non-zero term is a good approximation to
the potential for motion near the minimum.
Simple Harmonic Motion
The classic simple harmonic motion (SHM) example is a mass on a spring:
This is a one-dimensional problem with
F
kx
.
Newton’s 2
nd
law gives us:
2
0
0
mx
kx
x
x
(11.1)
where
2
0
k m
. We know the solutions to this equation already, but to aid us with the
more complex case of damped oscillations, we will review the mathematical technique
for solving such a second order linear differential equation.

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Equations of this type can be reduced by the substitution
rt
x
Be
. Substitution into
Equation (11.1) gives
2
2
0
0
rt
rt
r Be
Be
which leads to the
auxiliary equation
2
2
0
0
r
This is solved by simple algebra. The two roots are
0
r
i
The final solution to Equation (11.1) is a linear combination of these two solutions:
0
0
1
2
( )
i
t
i
t
x t
B e
B e
(11.2)
B
1
and
B
2
are complex, however, the physical quantity
x
(
t
)
must be real. A real
x
satisfies
x
x
(the “*” signifies complex conjugation:
i
i
).

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