dang (ttd73) – HW01 – Tsoi – (58020)
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001
(part 1 of 2) 10.0 points
This problem shows how dimensional analysis
helps us check and sometimes even find a
formula.
A rope has a cross section
A
= 11
.
4 m
2
and
density
ρ
= 2150 kg
/
m
3
. The “linear” density
of the rope
μ
, defined to be the mass per unit
length, can be written in the form
μ
=
ρ
x
A
y
.
Based on dimensional analysis, determine
the powers
x
and
y
by choosing an expression
below.
1.
μ
=
A
ρ
2.
μ
=
A
2
ρ
2
3.
μ
=
ρ
A
2
4.
μ
=
ρ
A
5.
μ
=
ρ A
correct
6.
μ
=
1
ρ A
7.
μ
=
A
ρ
2
8.
μ
=
ρ A
2
9.
μ
=
1
ρ A
2
10.
μ
=
A
2
ρ
Explanation:
Kilogram (kg): a unit of mass (
M
).
Meter (m): a unit of length (
L
).
[
x
] means ”the units of
x
”.
The units of both sides of any equation must
be the same for the equation to make sense.
The units of the left hand side (LHS) are
given as
[
μ
] =
M
L
=
ML

1
,
and the right hand side has
[
ρ
x
A
y
] =
parenleftbigg
M
L
3
parenrightbigg
x
×
(
L
2
)
y
=
M
x
L

3
x
L
2
y
=
M
x
L
2
y

3
x
,
thus
M
+1
L

1
=
M
x
L
2
y

3
x
.
The powers of the units of mass and length
need to be the same as for the LHS above, so
x
= 1
2
y
−
3
x
=
−
1
.
We can substitute the first equation into the
second one to obtain
y
= 1. Since
x
= 1 and
y
= 1, the answer is
μ
=
ρ
1
A
1
=
ρ A .
002
(part 2 of 2) 10.0 points
A simple pendulum is made out of a string
with length
L
and a mass
m
attached to one
end of the string. Its period
T
of oscillation
may depend on the gravitational acceleration
g
, and also depend on
L
and
m
.
Based on dimensional analysis, check which
one of the following expressions is dimension
ally acceptable, where
k
is a dimensionless
constant.
1.
T
=
k
radicalbigg
m g
L
2.
T
=
k
m L
g
3.
T
=
k
m g
L
4.
T
=
k
radicalBigg
L
g
correct
5.
T
=
k
g
L
6.
T
=
k
L
g
7.
T
=
k
radicalbigg
g
L
8.
T
=
k
radicalBigg
L
m g
Explanation:
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dang (ttd73) – HW01 – Tsoi – (58020)
2
Here we proceed in the same way.
A pe
riod is a measure of time, thus the correct
expression must have units of time.
bracketleftBigg
k
radicalBigg
L
g
bracketrightBigg
=
radicalBigg
L
L/T
2
=
T
is the correct one. As for the others,
bracketleftBig
k
m g
L
bracketrightBig
=
ML/T
2
L
=
MT

2
bracketleftbigg
k
m L
g
bracketrightbigg
=
ML
L/T
2
=
MT
2
bracketleftbigg
k
radicalbigg
m g
L
bracketrightbigg
=
radicalbigg
ML/T
2
L
=
M
1
2
T

1
bracketleftBigg
k
radicalBigg
L
m g
bracketrightBigg
=
radicalBigg
L
ML/T
2
=
M

1
2
T
bracketleftbigg
k
L
g
bracketrightbigg
=
L
L/T
2
=
T
2
bracketleftBig
k
g
L
bracketrightBig
=
L/T
2
L
=
T

2
bracketleftbigg
k
radicalbigg
g
L
bracketrightbigg
=
radicalbigg
L/T
2
L
=
T

1
So they are all incorrect, as they should be.
003
10.0 points
A flat circular plate of copper has a radius of
0
.
168 m and a mass of 68
.
4 kg.
What is the thickness of the plate?
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 Spring '08
 Kaplunovsky
 Acceleration, Velocity, Correct Answer, Orders of magnitude, m/s, Mg 3

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