Physics 325 Quiz #1 Solutions
Wednesday 27 September 2006
1a) In one dimension,
U
(
x
) =

Z
F
(
x
)
dx
=

A
k
sin
kx
+
C
It is the negative of the sine function, with a ﬁrst minimum at
x
=
π/
2
k
. The value of
the integration constant
C
does not change any result of the problem, and it can be set
to any number.
1b) All stable equilibrium points are of the form
x
n
= 2
nπ/k
+
π/
2
k
= (4
n
+ 1)
π/
2
k
,
where n is an integer,positive, negative, or zero. All unstable equilibrium points are of the
form
x
j
= 3
π/
2
k
+ 2
jπ/k
= (4
j
+ 3)
π/
2
k
, where
j
is a positive, negative, or zero integer.
1c) The frequency of small oscillations about each of the stable equilibrium points is the
same as any of the others because
U
(
x
) is periodic. Let’s pick the point at
x
0
=
π/
2
k
.
Let the deviation at
x
0
be
δx
, then we can do a Taylor expansion around the point
x
0
.
F
(
x
) =
A
cos
kx
0

Ak
sin(
kx
0
)
δx
+
...
≈ 
Akδx
By the equation for angular frequency in harmonic oscillation, we have
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 Fall '09
 Lamb
 Physics, mechanics, Energy, Force, Kinetic Energy, Potential Energy, 2m, 2k

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