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
00
0
2
1
2
x
x
dU
d U
Ux Ux
x x
dx
dx
=+
−
+
−
+
"
We are free to define
U(x
0
)=0
, and the first derivative vanishes by the condition that
x
0
be
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
2
x
dU
Ux
k
x
k
dx
≈≡
and
k
is positive since
x
0
is a minimum.
Simple harmonic motion is extremely important
because many potentials 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
ω
=
−→
+=
±±
(11.1)
where
2
0
km
≡
.
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.
Equations of this type can be reduced by the substitution
x=Be
rt
.
Substitution into
Equation (11.1) gives
22
0
0
rt
rt
rB
e
B
e
+
=
which leads to the
auxiliary equation
0
0
r
+
=

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*This is solved by simple algebra.
The two roots are
0
ri
ω
= ±
The final solution to Equation (11.1) is a linear combination of these two solutions:
( )
0
12
it
0
x
tB
e
B
e
−
=+
(11.2)
B
B
1
and
B
2
B
are complex, however, the physical quantity
x(t)
must be real.
A real
x
satisfies
x
*
=x
(the “*” signifies complex conjugation:
i
→
–i
), therefore
() ( )
00
0
**
*
1
2
0
x
e
B
e
x
e
B
e
ωω
−−
=
=
+
Or
*
B
B
=
So we really only have one arbitrary, complex, constant in this solution (but since it is
complex it has both a magnitude and a phase).

This is the end of the preview.
Sign up
to
access the rest of the document.