Physics 315:
Oscillations and Waves
Homework 2: Due in class on Wednesday, Sep. 16th
1. Demonstrate that in the limit
ν
→
2
ω
0
the solution to the damped har
monic oscillator equation becomes
x
(
t
) = (
x
0
+ [
v
0
+ (
ν/
2)
x
0
]
t
) e
−
ν t/
2
,
where
x
0
=
x
(0) and
v
0
= ˙
x
(0).
2. What are the resonant angular frequency and quality factor of the circuit
pictured below? What is the average power absorbed at resonance?
.
I
0
cos(
ω t
)
L
R
C
3. The power input
a
P
A
required to maintain a constant amplitude oscillation
in a driven damped harmonic oscillator can be calculated by recognizing
that this power is minus the average rate that work is done by the damping
force,
−
m ν
˙
x
.
(a) Using
x
=
x
0
cos
(
ω t
−
ϕ
), show that the average rate that the damping
force does work is
−
m ν ω
2
x
2
0
/
2.
(b) Substitute the value of
x
0
at an arbitrary driving frequency and, hence,
obtain an expression for
a
P
A
.
(c) Demonstrate that this expression yields Equation (3.56) of the lecture
notes in the limit that the driving frequency is close to the resonant
frequency.
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 Spring '08
 Staff
 Work, damped harmonic oscillator

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