Physics 315:
Oscillations and Waves
Homework 2: Solutions
1. The solution to the damped harmonic oscillator equation can be written
x
(
t
) =
x
0
e
−
ν t/
2
cos(
ω
1
t
) +
parenleftbigg
v
0
+
ν x
0
/
2
ω
1
parenrightbigg
e
−
ν t/
2
sin(
ω
1
t
)
,
where
ω
1
=
radicalbig
ω
2
0
−
ν
2
/
4. Here, it is assumed that
ν
≤
2
ω
0
. Furthermore,
x
0
=
x
(0) and
v
0
= ˙
x
(0). Taking the limit
ν
→
2
ω
0
, which is equivalent to
taking the limit
ω
1
→
0, we find that
x
(
t
)
→
x
0
e
−
ν t/
2
+
parenleftbigg
v
0
+
ν x
0
/
2
ω
1
parenrightbigg
e
−
ν t/
2
ω
1
t,
since cos
θ
≃
1 and sin
θ
≃
θ
when

θ
 ≪
1. Hence,
x
(
t
)
→
(
x
0
+ [
v
0
+ (
ν/
2)
x
0
]
t
) e
−
ν t/
2
.
2. Let
I
1
(
t
),
I
2
(
t
), and
I
3
(
t
) be the currents flowing in the left, middle, and
right legs of the circuit, respectively. If
I
(
t
) =
I
0
cos(
ω t
) is the current fed
into the circuit then conservation of current requires that
I
(
t
) =
I
1
(
t
) +
I
2
(
t
) +
I
3
(
t
)
.
Since the three legs of the circuit are connected in parallel, the potential
drops across them are the same. The potential drop across the left leg is
L
dI
1
dt
,
whereas the potential drop across the middle leg is
R I
2
,
and the potential drop across the right leg is
integraldisplay
t
0
I
3
(
t
′
)
dt
′
slashbigg
C.
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Hence, the common potential drop across all three legs is
V
(
t
) =
L
dI
1
dt
=
R I
2
=
integraldisplay
t
0
I
3
(
t
′
)
dt
′
slashbigg
C.
It follows that
I
2
=
L
R
dI
1
dt
,
and
L C
d
2
I
1
dt
2
=
I
3
=
I
−
I
1
−
I
2
=
I
−
I
1
−
L
R
dI
1
dt
.
Thus,
d
2
I
1
dt
2
+
ν
dI
1
dt
+
ω
2
0
I
1
=
ω
2
0
I
0
cos(
ω t
)
,
(1)
where
ω
0
= 1
/
√
L C
and
ν
= 1
/R C
. This is the driven damped harmonic
oscillator equation. The resonant frequency is
ω
0
=
1
√
L C
,
whereas the quality factor takes the form
Q
f
=
ω
0
ν
=
R
radicalbig
L/C
.
At the resonant frequency (
ω
=
ω
0
), the first and third terms on the left
hand side of Eq. (1) cancel (since
d
2
/dt
2
≡ −
ω
2
), and we are left with
ν
dI
1
dt
=
ω
2
0
I
0
cos(
ω
0
t
)
.
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 Spring '08
 Staff
 Work, Cos, potential drop, Eqs., harmonic oscillator equation

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