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CHAPTER 6
Fundamental Dimensions and Units
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-1
Mars Climate Orbiter
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Mars Climate Orbiter
Source: NASA JPL (http://mars.jpl.nasa.gov/)
Material to be Covered
Chapter 6: Sections 1 – 6
Outline
In this chapter we will
• Explain fundamental dimensions and units
• Explain the steps necessary to convert
information from one system of units to
another
• Emphasize the importance of showing
appropriate units with all calculations.
• Discuss how you can learn the engineering
fundamental concepts using fundamental
dimensions
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-5
Fundamental Dimensions
The physical quantities used to fully describe
natural events and our surroundings are:
• Length
• Temperature
• Mass
• Amount of substance
• Time
• Luminous intensity
• Electric current
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-6
Units
• Used to measure physical dimensions
• Appropriate divisions of physical dimensions
to keep numbers manageable
 19 years old instead of 612,000,000
seconds old
• Common systems of units
 International System (SI) of Units
 British Gravitational (BG) System of Units
 U.S. Customary Units
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-7
Units – SI
• Most common system of units used in the
world
• Examples of SI units are: kg, N, m, cm,
• Approved by the General Conference on
Weights and Measures (CGPM)
• Series of prefixes & symbols of decimal
multiples (adapted by CGPM, 1960)
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-8
Fundamental Unit of Length
meter (m) – length of the path traveled
by light in a vacuum during a time
interval of 1/299,792,458 of a second
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6-9
Fundamental Unit of Mass
kilogram (kg) – a unit of mass; it is equal to
the mass of the international prototype of
the kilogram
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6-10
Fundamental Unit of Time
second (s) – duration of 9,192,631,770
periods of the radiation corresponding to the
transition between the 2 hyperfine levels of
the ground state of cesium 133 atom
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-11
Fundamental Unit of Electric Current
ampere (A) – constant current which, if
maintained in 2 straight parallel conductors
of infinite length, of negligible circular cross
section, and placed 1 meter apart in a
vacuum, would produce between these
conductors a force equal to 2x10-7N/m length
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-12
Fundamental Unit of Temperature
kelvin (K) – unit of thermodynamic
temperature, is the fraction 1/273.16 of
thermodynamic temperature of the triple
point of water
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6-13
Fundamental Unit of Amount of
Substance
mole (mol) – the amount of substance of a
system that contains as many elementary
entities as there are atoms in 0.012 kilogram of
carbon 12
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-14
Fundamental Unit of Luminous Intensity
candela (cd) – in a given direction, of a
source that emits monochromatic radiation
of frequency 540x1012 hertz and that has a
radiant intensity in that direction of 1/683
watt per steradian
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-15
SI – Prefix & Symbol
Adopted by
CGPM in 1960
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6-16
Examples of Derived Units in Engineering
More in
chapters
7-13
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6-17
British Gravitation (BG) System
More on
temperature
in chapter 11
• Primary units are
 foot (ft) for length (1 ft = 0.3048 m)
 second for time
 pound (lb) for force (1 lb = 4.448 N)
o
 Fahrenheit ( F) for temperature
• Slug is unit of mass which is derived from
Newton’s second law
2
 1 lb = (1 slug)(1 ft/s )
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-18
U.S. Customary System of Units
More on
derived units
in chapters
7-13
• Primary units are
 Foot (ft) for length (1 ft = 0.3048 m)
 second for time
 pound mass (lbm) for mass (1 lbm =
0.453592 kg, 1slug = 32.2 lbm)
• Pound force (lbf) is defined as the weight of
an object having a mass of 1 lbm at sea level
and at a latitude of 45o, where acceleration
due to gravity is 32.2 ft/s2 (1lbf = 4.448 N)
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-19
Examples of SI Units in Everyday Use
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6-20
Examples of U.S. Customary Units in
Everyday Use
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-21
Systems of Units and Conversion Factors
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6-22
Unit Conversion
• In engineering analysis and design, there may be
a need to convert from one system of units to
another
• When communicating with engineers outside of
U.S.
• Important to learn to convert information from
one system of units to another correctly
• Always show the appropriate units that go with
your calculations
• See the conversion tables given in the book
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-23
Example 6.1 – Unit Conversion
Given: a person who is 6’-1” tall and weighs 185
pound force (lbf)
Find: height and weight in SI units
Solution:
Person'
sheight,
H

.3048
m
 1ft0

H
.854
m



1
6ft1in

in
12
 1ft 

Person'
s weight,
W
4

.448
N

W185
lb
.88
N
f
 1lb 
822
f


© 2011 Cengage Learning Engineering. All Rights Reserved.
6-24
Example 6.1 – Unit Conversion
Given: a person is driving a car at a speed of 65
miles per hour (mph) over a distance of 25 miles
Find: speed and distance in SI units
Solution:
Speed
of
car,
S
ft
m
 miles

5280

0.3048

S
104
,607
m/h

104.607
km/h
65 




h
1
mile
1
ft






m
h


1
or
S
104
,607
.057
m/s



29
h
s

3600
Distance
traveled,
D
5280
ft
m
km

0.3048

1

25

D

miles
.233
km





40
mile
ft 
m
1

 1
1000

© 2011 Cengage Learning Engineering. All Rights Reserved.
6-25
Example 6.1 – Unit Conversion
Given: outside temperature is 80oF and has a
density of 0.0735 pound mass per cubic foot (lbm/ft3)
Find: temperature and density in SI units
Solution:
Temperatur
e
of
air,
T
5
5
C
 T
F

26
 80
T
32

32
.7

C
9
9
Density
of
air,

0

lb
.453
kg
ft 


 1
3
m



0
.0735

1
.
176
kg/m




3 
0.3048
ft
1
lb
m




m 

3
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-26
Example 6.2 – Unit Conversion
Given: area A = 100 cm2
Find: A in m2
Solution:


2
m
1

2
A

100
cm

0
.
01
m


100
cm


2
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-27
Example 6.2 – Unit Conversion
Given: volume V = 1000 mm3
Find: V in m3
Solution:


3
m
1
 
63
V

1000
mm

10
m


1000
mm


3
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-28
Example 6.2 – Unit Conversion
Given: atmosphere pressure P = 105 N/m2
Find: P in lb/in2
Solution:
2
N
1
lb
0.0254
m






5
2
P

10

14
.
5
lb/i
 2





4.448
N
1
in
m





© 2011 Cengage Learning Engineering. All Rights Reserved.
6-29
Example 6.2 – Unit Conversion
Given: density of water = 1000 kg/m3
Find: density in lbm/ft3
Solution:



1
lb
1
m
 kg



3
m



1000

62
.
5
lb
/ft
 3

 
m


m
0.4536
kg
3.28
ft


 


© 2011 Cengage Learning Engineering. All Rights Reserved.
3
6-30
Examples of Unit Conversion
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6-31
Dimension Homogeneity
• Given: L = a + b + c
• Left hand side of equation should have the
same dimension as right hand side of
equation
• If L represents dimension length, then a, b,
and c must also have the dimension of
length – this is called dimensionally
homogeneous
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-32
Example 6.3 – Dimension Homogeneity
Given:
PL
d
AE
where
d = end deflection, in m
P = applied load, in N
L = length of bar, in m
A = cross-sectional area of bar, in m2
E: modulus of elasticity
Find: units for E
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-33
Example 6.3 – Dimension Homogeneity
Solution:




PL
N
m
d
m
 2
AE
mE




N
m
N
E
 2
2
m
m m
 
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-34
Example 6.3 – Dimension Homogeneity
T
1T
2
Given: qkA
L
Where
q = heat transfer rate
k = thermal conductivity of the solid
material in W/m●°C
A = area in m2
T1 – T2 = temperature difference, °C
L = thickness of the material, m
Find: units for q
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-35
Example 6.3 – Dimension Homogeneity
Solution:



T

T
W
C
2
1
2

q

kA

m

W



L
m
m

C
 
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-36
Numerical versus Symbolic Solutions
• For problems requiring a numerical solution,
data is given and calculations are carried out.
• For problems requiring a symbolic solution,
the steps and final answer are presented in
terms of the variables.
Example 6.5 – Numerical Versus
Symbolic Solutions
Given: hydraulic system
shown
Find: m2
Numerical solution:
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-38
Example 6.5 – Numerical Versus
Symbolic Solutions
Given: hydraulic system
shown
Find: m2
Numerical solution:


2
F

m
g

100
kg
9
.
81
m/s

981
N
1
1


A

0
.
15
m
2


F
F

981
N
8829
N
2
1
2
A



0
.
05
m
1
2


2
F

m
9
.
81
m/s

8829
Nm

900
kg
2
2
2
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-39
Example 6.5 – Numerical Versus
Symbolic Solutions
Given: hydraulic system
shown
Find: m2
Symbolic solution:
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6-40
Example 6.5 – Numerical Versus
Symbolic Solutions
Given: hydraulic system
shown
Find: m2
Symbolic solution:
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6-41
Significant Digits (Figures)
• Engineers make measurements and carry
out calculations
• Engineers record the results of
measurements and calculations using
numbers.
• Significant digits (figures) represent
(convey) the extend to which recorded or
computed data is dependable.
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6-42
Significant Digits – How to Record a
Measurement
Least count – one half
of the smallest scale
division
What should we
record for this
temperature
measurement?
71 ± 1oF
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6-43
Significant Digits – How to Record a
Measurement
What should we record for the length?
3.35 ± 0.05 in.
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6-44
Significant Digits – How to Record a
Measurement
What should we record for this pressure?
7.5 ± 0.5 in.
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6-45
Significant Digits
• 175, 25.5, 1.85, and 0.00125 each has three
significant digits.
• The number of significant digits for the
number 1500 is not clear.
 It could be 2, 3, or 4
3
2
 If recorded as 1.5 x 10 or 15 x 10 ,
then 2 significant digits
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-46
Significant Digits – Addition And
Subtraction Rules
When adding or subtracting numbers, the result of the
calculation should be recorded with the last significant digit in
the result determined by the position of the last column of digits
common to all of the numbers being added or subtracted.
For example,
152.47
+
3.9
or
132. 853
-
5
156.37
127.853 (your calculator will display)
156.3
127
© 2011 Cengage Learning Engineering. All Rights Reserved.
(however, the result should be
recorded this way)
6-47
Significant Digits – Multiplication and
Division Rules
When multiplying or dividing numbers, the result of the
calculation should be recorded with the least number of
significant digits given by any of the numbers used in the
calculation.
For example,
152.47
×
3.9
or
152.47
÷
3.9
594.633
39.0948717949 (your calculator will display)
5.9 x 102
39
© 2011 Cengage Learning Engineering. All Rights Reserved.
(however, the result should be
recorded this way)
6-48
Significant Digits – Examples
276.34
+ 12.782
289.12
2955
x 326
9.63 x 105
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6-49
Rounding Numbers
In many engineering calculations, it may
be sufficient to record the results of a
calculation to a fewer number of
significant digits than obtained from the
rules we just explained
56.341 to 56.34
12852 to 1.285 x 104
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-50
Summary
• You should understand the importance of
fundamental dimensions in engineering
analysis
• You should know the most common
systems of units
• You should know how to convert values
from one system of units to another
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6-51
Summary
• You should understand the difference
between numerical and symbolic solutions
• You should know how to present the result
of your calculation or measurement using
correct number of significant digits
© 2011 Cengage Learning Engineering. All Rights Reserved.
6-52
Units
Questions?