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Homework Assignment #2 Solutions  1
ECE 473/TAM 413
Homework Assignment #2 Solutions
1. The tension in a string is provided by hanging a 3kg mass at one end; the other end is rigidly
fixed. The length of the string is 2.5 m and its mass is 50 g. Determine the phase speed of
waves on the string.
Tension in the string is T = F = mg = (3 kg)(9.81 m/s
2
) = 29.4 N
Mass per unit length is
!
L
=
m
L
=
0.05 kg
2.5 m
=
0.02 kg /m
Phase speed is
c
=
T
!
L
=
29.4 N
0.02 kg / m
=
38.3m /s
2. Problem 2.3.1 in Kinsler et al. (show all work). Put your final answers in the form of a wave
equation.
(a) Assume the linear density varies with position, that is,
!
L
=
!
L
x
( )
.
The tension force on each element is given by:
df
y
=
!
!
x
T
!
y
!
x
"
#
$
%
&
’
dx
=
T
!
2
y
!
x
2
dx
(see Eq. 2.3.4)
Using Newton’s 2nd Law:
f
y
=
ma
=
m
!
2
y
!
t
2
, but the mass for an element is:
dm
=
!
L
x
( )
dx
for
some element at x, giving
df
y
=
dm
( )
a
=
!
L
x
( )
"
2
y
"
t
2
dx
.
Equating forces
T
!
2
y
!
x
2
dx
=
"
L
x
( )
!
2
y
!
t
2
dx
and rearranging yields:
!
2
y
!
t
2
=
T
"
L
x
( )
!
2
y
!
x
2
.
(b) Assume the string hangs vertically supported only at the upper end. Therefore, the tension is
given by the force of gravity and the mass of the string hanging below the position x for a dx
element at x. Thus, the tension at x is given by:
T
=
m x
( )
g
=
!
L
xg.
From (2.3.4),
df
y
=
!
!
x
T
!
y
!
x
"
#
$
%
&
’
dx
=
!
!
x
(
L
xg
!
y
!
x
"
#
$
%
&
’
dx
=
(
L
g
!
!
x
x
!
y
!
x
"
#
$
%
&
’
dx
Using Newton’s 2nd Law:
df
y
=
dm
( )
a
=
!
L
"
2
y
"
t
2
dx
, where
dm
=
!
L
dx
.
Equating forces
!
L
g
"
"
x
x
"
y
"
x
#
$
%
&
’
(
dx
=
!
L
"
2
y
"
t
2
dx
and rearranging yields:
!
2
y
!
t
2
=
g
!
!
x
x
!
y
!
x
"
#
$
%
&
’
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View Full DocumentHomework Assignment #2 Solutions  2
3. Problem 2.5.1 in Kinsler et al.
y x,t
( )
moves 5 cm per second, that is, c = 5 cm/s
4. Given a finitelength string driven at x = 0 under steadystate conditions by
Fe
j
!
t
and
supported rigidly at the other end at x = L, determine (a) the instantaneous input power
!
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 Fall '08
 OBRIAN

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