sanchez (ds28677) – homework 33 – Turner – (58220)
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001
(part 1 of 2) 10.0 points
A standing wave of frequency 5 hertz is set up
on a string 2 meters long with nodes at both
ends and in the center, as shown.
2 meters
Find the speed at which waves propagate
on the string.
1.
2
.
5 m
/
s
2.
20 m
/
s
3.
10 m
/
s
correct
4.
0
.
4 m
/
s
5.
5 m
/
s
Explanation:
Let :
f
= 5 Hz
and
λ
= 2 m
.
The wavelength is
λ
= 2 m, so the wave
speed is

vectorv

=
f λ
= (5 Hz)(2 m) =
10 m/s
.
002
(part 2 of 2) 10.0 points
Find the fundamental frequency of vibration
of the string.
1.
2
.
5 Hz
correct
2.
10 Hz
3.
1 Hz
4.
7
.
5 Hz
5.
5 Hz
Explanation:
2 meters
The fundamental wave has only two nodes
at the ends, so its wavelength is
λ
= 4 m and
the fundamental frequency is
f
=
v
λ
=
10 m
/
s
4 m
=
2.5 Hz
.
003
10.0 points
Two wires are made of the same material but
the second wire has twice the diameter and
twice the length of the first wire.
When the
two wires are stretched, and the tension in
the second wire is also twice the tension in the
first wire, the fundamental frequency of the
first wire is 670 Hz.
What is the fundamental frequency of the
second wire?
Correct answer: 236
.
881 Hz.
Explanation:
Let :
f
0
1
= 670 Hz
.
The second wire has twice the radius and
hence four times the cross sectional area (=
π r
2
) of the first wire.
Since the two wires
are made from the same material, the linear
density
μ
=
π d
2
4
ρ
of the second wire is four times that of the
first:
μ
2
= 4
μ
1
.
The speed of transverse waves in a string or
a wire is
v
=
radicalBigg
T
μ
, and since the second wire
has twice the tension of the first wire,
v
2
=
radicalBigg
T
2
μ
2
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sanchez (ds28677) – homework 33 – Turner – (58220)
2
=
radicalBigg
2
T
1
4
μ
1
=
v
1
radicalbigg
1
2
.
The fundamental frequency of standing waves
in the wire is given by the condition
L
=
λ
2
,
so
f
0
=
v
λ
=
v
2
L
.
For the two wires,
v
2
=
v
1
√
2
and
L
2
= 2
L
1
, so
f
0
2
=
f
0
1
2
√
2
=
670 Hz
2
√
2
= 236
.
881 Hz
.
004
10.0 points
A nylon guitar string vibrates in a standing
wave pattern shown below.
1
.
8 m
What is the wavelength of the wave?
Correct answer: 1
.
2 m.
Explanation:
Let :
l
= 1
.
8 m
.
The wavelength is the length of two loops:
λ
=
2
l
3
=
2 (1
.
8 m)
3
=
1
.
2 m
.
005
(part 1 of 2) 10.0 points
Consider a vibrating piano string. The string
is under a tension
T
, has a length
L
and
diameter
d
.
What is the wavelength of its third har
monic?
1.
λ
3
=
L
2.
λ
3
=
d
3
3.
λ
3
=
3
L
2
4.
λ
3
=
L
6
5.
λ
3
= 3
L
6.
λ
3
=
2
L
3
correct
7.
λ
3
=
L
2
8.
λ
3
= 6
L
9.
λ
3
=
L
3
10.
λ
3
= 3
√
d L
Explanation:
Basic Concepts
For a string fixed at both
ends, the normal modes have a wavelength
λ
n
=
2
L
n
.
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 Spring '08
 Turner
 Physics, Work, Light, Wavelength, Sanchez, Standing wave

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