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ASSIGNMENT 2
Physics 218
Due Sep 10
Fall 2004
1. Consider the piano string of Prob. 2 in Assignment 1. That is, consider a steel piano
wire
D
= 1 mm in diameter,
`
=0
.
604 m long, with
ρ
= 8 g/cm
3
, and stretched with
tension
τ
0
= 628 N, so it is tuned to middle C (261.6 Hz). If the string is oscillating in
only its fundamental mode with
A
= 2 mm, what is the total energy of vibration?
2. Derive Eqs. (1.8.28), the orthogonality relations for sin(
ω
n
t
) and cos(
ω
n
t
). Recall that
T
1
is the period of the fundamental,
i.e.
,
T
1
=1
/f
1
=2
π/ω
1
.
3. (a) Calculate
∂η
(
x, t
)
/∂t
for the Fourier series in Eq. (1.8.24).
(b) Use the
∂η
(
x, t
)
/∂t
that you calculated in Part (a) to derive Eq. (1.8.30) for the
average kinetic energy of a vibrating string ±xed at
x
= 0 and
x
=
`
:
h
K
i
=
`λ
0
8
∞
X
n
=1
ω
2
n
(
a
2
n
+
b
2
n
)
(c) Combine Eqs. (1.8.29) and (1.8.30) to obtain the average total energy
h
E
i
in Eq. (1.8.31)
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 Fall '04
 POLLACK
 Physics, Magnetism

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