Dimensional analysis problems
Math 647.600
1. Why do stringed musical instruments have strengths of di erent lengths
and thicknesses? Assume that the fundamental frequency
ω
of vibration
of a string depends on its length
l
, mass per unit length
μ
, and tension
(force)
F
on the string. Prove that
ω
must be proportional to
p
F/μ
l
.
Answer:
The physical quantities involved are frequency
ω
, mass per unit
length
μ
, length
l
, and tension
F
. The dimensions of these are:
[
ω
] =
T

1
,
[
μ
] =
ML

1
,
[
l
] =
L
, and
[
F
] =
MLT

2
, where the dimensions of
frequency and force are given in the class notes. A product of these is
Π =
ω
a
μ
b
l
c
F
d
, which has dimensions
[Π]
=
T

a
(
ML

1
)
b
L
c
(
MLT

2
)
d
=
T

a

2
d
M
b
+
d
L

b
+
c
+
d
.
This will be dimensionless if and only if

a

2
d
=
0
b
+
d
=
0

b
+
c
+
d
=
0
.
The augmented matrix for this system is

1
0
0

2
.
.
.
0
0
1
0
1
.
.
.
0
0

1
1
1
.
.
.
0
,
which row reduces to
1
0
0
2
.
.
.
0
0
1
0
1
.
.
.
0
0
0
1
2
.
.
.
0
.
The interpretation of this is that
d
is arbitrary,
c
=

2
d
,
b
=

d
, and
a
=

2
d
. Thus the only possibility for a dimensionless
Π
is
Π =
(
ω

2
μ

1
l

2
F
)
d
.
The arbitrary power
d
may be ignored, since a function of
(
ω

2
μ

1
l

2
F
)
d
is also a function of
ω

2
μ

1
l

2
F
. Thus a physical law relating these
quantities must be of the form
f
(
ω

2
μ

1
l

2
F
)
= 0
,
1
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View Full Documentwhich we may generically solve for
Π
to get
ω

2
μ

1
l

2
F
=
k,
for some dimensionless constant
k
. Thus
ω
=
s
kF
μl
2
=
k
±
p
F/μ
l
,
as desired.
2. We now want to include frictional e ects and the initial angle in analyzing
pendulums. As in the lecture notes, the length of the pendulum is
l
and
its mass is
m
.
(a) Suppose that the frictional force is due primarily to air and is pro
portional to
v
2
with constant of proportionality
k
. Let
τ
be the
time required for the pendulum to reach half its initial amplitude
θ
.
Determine the dimensions of
k
. Show that
τ
=
s
l
g
G
±
θ,
kl
m
¶
for some function
G
.
Answer:
The quantities involved are
τ
, length
l
, mass
m
, initial
amplitude
θ
, acceleration due to gravity
g
, and the drag constant
k
.
The dimensions of all but the last quantity are clear. If drag force
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 Spring '08
 Howard
 Math, Mass, Row echelon form, θ, cooking time, physical law

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