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
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Version 012 – Exam1 – Chiu – (60180)
2
This is
not
dimensionally consistent.
(4)
v
2
= 2
a s
+
k s v
t
[
v
2
] =
L
2
T
−
2
bracketleftbigg
2
a s
+
k s v
t
bracketrightbigg
=
LT
−
2
L
+
LLT
−
1
T
−
1
=
L
2
T
−
2
This is also consistent.
(5)
s
=
s
0
+
v t
+
v
2
a
[
s
] =
L
bracketleftbigg
s
0
+
vt
+
v
2
a
bracketrightbigg
=
L
+
LT
−
1
T
+
L
2
T
−
2
LT
−
2
=
L
This is also consistent.
So only
t
=
k
radicalbigg
s
g
+
a
v
is dimensionally
incorrect
.
003
10.0 points
A newly discovered giant planet has an aver
age radius 19
.
999 times that of the Earth and
a mass 985 times that of the Earth.
Calculate the ratio of the new planet’s den
sity to the Earth’s density.
1. 0.301749
2. 0.123143
3. 0.0644409
4. 0.196148
5. 0.106891
6. 0.806
7. 0.0354938
8. 0.0632716
9. 0.0268675
10. 0.251707
Correct answer: 0
.
123143.
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 Spring '08
 Turner
 Acceleration, Velocity, Correct Answer, tQ tR tS

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