Version 012 – Exam1 – Chiu – (60180)
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001
10.0 points
The velocity of a transverse wave traveling
along a string depends on the tension
F
=
m a
of the string and its mass per unit length
μ
.
Assume
v
=
F
x
μ
y
.
The powers of
x
and
y
may be determined based on dimensional
analysis.
By equating the powers of mass,
length, and time, one arrives correspondingly
at a set of three equations.
Choose the correct expressions for
x
and
y
.
1.
0 =
x
+
y,
1 =
x
+
y,
−
1 =
−
2
x
2.
0 =
x
+
y,
1 =
x
−
y,
1 =
−
2
x
3.
0 =
x
+
y,
1 =
x
−
y,
−
1 =
−
2
x
correct
4.
0 =
x
−
y,
1 =
x
−
y,
−
1 =
−
2
x
5.
0 =
x
−
y,
1 =
x
−
y,
1 =
−
2
x
6.
0 =
x
−
y,
1 =
x
+
y,
−
1 =
−
2
x
7.
0 =
x
+
y,
1 =
x
+
y,
1 =
−
2
x
8.
0 =
x
+
y,
0 =
x
−
y,
1 =
−
2
x
9.
1 =
x
+
y,
1 =
x
−
y,
1 =
−
2
x
10.
0 =
x
−
y,
1 =
x
+
y,
1 =
−
2
x
Explanation:
Recall that the dimensions of velocity, ten-
sion, and mass per unit length are
[
v
] =
L
T
,
[
F
] =
ML
T
2
,
[
μ
] =
M
L
hence
L
T
= [
F
x
μ
y
] =
parenleftbigg
ML
T
2
parenrightbigg
x
parenleftbigg
M
L
parenrightbigg
y
=
M
x
+
y
L
x
−
y
T
−
2
x
Equating the powers of
M
yields 0 =
x
+
y
.
Equating the powers of
L
yields 1 =
x
−
y
.
Equating the powers of
T
yields
−
1 =
−
2
x
.
002
10.0 points
Consider the following set of equations, where
s
,
s
0
,
x
and
r
have units of length,
t
has units
of time,
v
has units of velocity,
g
and
a
have
units of acceleration, and
k
is dimensionless.
Which one is dimensionally
incorrect
?
1.
s
=
s
0
+
v t
+
v
2
a
2.
t
=
k
radicalbigg
s
g
+
a
v
correct
3.
t
=
v
a
+
x
v
4.
v
2
= 2
a s
+
k s v
t
5.
a
=
g
+
k v
t
+
v
2
s
0
Explanation:
For an equation to be dimensionally cor-
rect, all its terms must have the same units.
(1)
t
=
v
a
+
x
v
[
t
] =
T
bracketleftBig
v
a
bracketrightBig
+
bracketleftBig
x
v
bracketrightBig
=
LT
−
1
LT
−
2
+
L
LT
−
1
=
T
+
T
=
T
It is consistent.
(2)
a
=
g
+
k v
t
+
v
2
s
0
[
a
] =
LT
−
2
bracketleftbigg
g
+
kv
t
+
v
2
s
0
bracketrightbigg
=
LT
−
2
+
LT
−
1
T
+
L
2
T
−
2
L
=
LT
−
2
It is also consistent.
(3)
t
=
k
radicalbigg
s
g
+
a
v
[
t
] =
T
bracketleftbigg
k
radicalbigg
s
g
+
a
v
bracketrightbigg
=
radicalbigg
L
LT
−
2
+
LT
−
2
LT
−
1
=
T
+
T
−
1