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CHAPTER
1
INTRODUCTION AND
MATHEMATICAL CONCEPTS
CONCEPTUAL QUESTIONS
____________________________________________________________________________________________
1.
REASONING AND SOLUTION
The quantity tan
θ
is dimensionless and has no units.
The units of the ratio
x/v
are
m
(m / s)
m
s
m
s
=
F
H
G
I
K
J
=
Thus, the units on the left side of the equation are not consistent with those on the right side,
and the equation tan
=
x
/
v
is not a possible relationship between the variables
x
,
v
, and
.
____________________________________________________________________________________________
2.
SSM
REASONING AND SOLUTION
It is not always possible to add two numbers that
have the same dimensions.
In order to add any two physical quantities they must be
expressed in the same
units
.
Consider the two lengths:
1.00 m and 1.00 cm.
Both
quantities are lengths and, therefore, have the dimension [L].
Since the units are different,
however, these two numbers cannot be added.
____________________________________________________________________________________________
3.
REASONING AND SOLUTION
a. The SI unit for
x
is m.
The SI units for the quantity
vt
are
m
s
⎛
⎝
⎜
⎞
⎠
⎟
(s)
=
m
m
s
(s)
m
F
H
G
I
K
J
=
Therefore, the units on the left hand side of the equation are consistent with the units on the
right hand side.
b. As described in part
a
, the SI units for the quantities
x
and
vt
are both m.
The SI units for
the quantity
1
2
at
2
are
m
s
(s )
m
2
2
F
H
G
I
K
J
=
Therefore, the units on the left hand side of the equation are consistent with the units on the
right hand side.
c. The SI unit for
v
is m/s.
The SI unit for the quantity
at
is
m
s
(s)
m
s
2
F
H
G
I
K
J
=
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INTRODUCTION AND MATHEMATICAL CONCEPTS
Therefore, the units on the left hand side of the equation are consistent with the units on the
right hand side.
d. As described in part
c
, the SI units of the quantities
v
and
at
are both m/s.
The SI unit of
the quantity
1
2
at
3
is
m
s
(s )
m s
2
3
F
H
G
I
K
J
=⋅
Thus, the units on the left hand side are
not
consistent with the units on the right hand side.
In fact, the right hand side is not a valid operation because it is not possible to add physical
quantities that have different units.
e. The SI unit for the quantity
v
3
is m
3
/s
3
.
The SI unit for the quantity 2
ax
2
is
m
s
(m )
m
s
2
2
3
2
F
H
G
I
K
J
=
Therefore, the units on the left hand side of the equation are
not
consistent with the units on
the right hand side.
f. The SI unit for the quantity
t
is s.
The SI unit for the quantity
2
x
a
is
m
(m / s )
m
s
m
ss
2
2
2
=
F
H
G
I
K
J
==
Therefore, the units on the left hand side of the equation are consistent with the units on the
right hand side.
____________________________________________________________________________________________
4.
REASONING AND SOLUTION
a. The dimension of a physical quantity describes the physical
nature
of the quantity and
the
kind
of unit that is used to express the quantity.
It is possible for two quantities to have
the same dimensions but different units.
All lengths, for example, have the dimension [L].
However, a length may be expressed in any length unit, such as kilometers, meters,
centimeters, millimeters, inches, feet or yards.
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This note was uploaded on 05/09/2008 for the course PHYS 23 taught by Professor Holland during the Spring '08 term at Pacific.
 Spring '08
 holland

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