Dimensional Analysis Definitions

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Dimensional AnalysisDefinitions
SWTJC STEM ENGR 1201
Dimension A physical property being measured". 
Qualitative Asks what?
Unit Addresses the quantitative aspect of a
dimension". 
Quantitative How much?
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Dimensional AnalysisExamples
SWTJC STEM ENGR 1201
Dimension Symbol Unit Example
Length L metersfoot Length of room Stopping distance
Area L2 meters2foot2 Area of room floor Cross-section of I-beam
Time T seconds Stopping time of car Flight time of arrow
Mass M kilogramsslug Mass of engine block Mass of electron
Force F newtonspounds Weight of rocketRocket engine thrust
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Dimensional AnalysisDimension Types
SWTJC STEM ENGR 1201
Dimensions are classified as one of three types
(1) fundamental (2) supplementary (3)
derived
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Dimensional AnalysisFundamental Dimensions
SWTJC STEM ENGR 1201
Fundamental dimensions Certain fundamental
qualities such as length, mass, force, and time
are symbolized with a single letter. 
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Dimensional Analysis Fundamental Dimensions
Dimensional Systems
SWTJC STEM ENGR 1201
Dimensional System Minimum set of fundamental
dimensions and associated base units that cover
all needed physical properties for a field of
science or engineering. Seven such systems
generally recognized internationally.  See table
14.1 on page 365 of the text.  Only two are
used extensively, (1) SI (Systeme
International) Metric System (2) USCS
(United States Customary System)
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Dimensional Analysis Fundamental Dimensions SI
(Metric)
SWTJC STEM ENGR 1201
SI - Systeme International or Metric System
Note Force and charge are not fundamental
dimensions.
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Dimensional Analysis Fundamental Dimensions USCS
SWTJC STEM ENGR 1201
USCS - United States Customary System
Note Mass and charge are not fundamental
dimensions.
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Dimensional Analysis Fundamental Dimensions
Absolute vs. Gravitational
SWTJC STEM ENGR 1201
SI (Systeme International)         

1. In the USCS system, force is a fundamental
   dimension.  A standard weight is involved in the
   definition of force tying the USCS system to
   gravitational effects.  The USCS system is called
   a gravitational system. For earthbound problems
   this fine, but for space mechanics it presents
   difficulties.
2. In the SI system, mass is a fundamental dimension
   making the entire system independent of
   gravitational considerations.  For this reason,
   the SI system is called an absolute system. It
   works everywhere!

1 In SI, force is derived from mass (F M.L/T2)
making the force unit a kg.m/s2 that is called a
newton (N).2 The use of the letter "A" to refer
to both the dimension and the unit is a bit
confusing.  You will have to rely on the context
of its use to determine which it is.
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Dimensional AnalysisTypes of DimensionsSupplemen
tary
SWTJC STEM ENGR 1201
Dimensions are classified as one of three types
(1) fundamental
(2) supplementary
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Dimensional Analysis Supplementary Dimensions
SWTJC STEM ENGR 1201
SI (Systeme International)         
Supplementary dimensions Embody geometric
concepts needed in the mathematical formulation
of natural laws". 
1 In SI, force is derived from mass (F M.L/T2)
making the force unit a kg.m/s2 that is called a
newton (N).2 The use of the letter "A" to refer
to both the dimension and the unit is a bit
confusing.  You will have to rely on the context
of its use to determine which it is.
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Dimensional AnalysisTypes of DimensionsDerived
SWTJC STEM ENGR 1201
Dimensions are classified as one of three types
(1) fundamental (2) supplementary
(3) derived
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Dimensional AnalysisTypes of DimensionsDerived
SWTJC STEM ENGR 1201
Derived dimensions Associated with physical
properties that can be written as some
combination of fundamental dimensions". 

1. Fundamental dimension Length L Base unit m
   (meters)Derived dimension Area L2 Base unit
   m2 (meters squared)

Area L . L L2 4 m2
L
L
Length 2 m
L
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Dimensional AnalysisTypes of DimensionsDerived
SWTJC STEM ENGR 1201
Derived dimensions Associated with physical
properties that can be written as some
combination of fundamental dimensions". 
2. Fundamental dimension Force F Base unit
lb (pound)Derived dimension Pressure (force
per unit of area) F/L2 Base unit lb/ft2
(pounds per foot squared)
Pressure Force /Area F/L2 lb/ft2
F (lb) (Distributed)
Force F lb (pounds)
L (ft)
L (ft)
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Dimensional Analysis Derived DimensionsSI
SWTJC STEM ENGR 1201
2
2
2
2
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Dimensional Analysis Derived DimensionsUSCS
SWTJC STEM ENGR 1201
SI (Systeme International)         
1 In SI, force is derived from mass (F M.L/T2)
making the force unit a kg.m/s2 that is called a
newton (N).2 The use of the letter "A" to refer
to both the dimension and the unit is a bit
confusing.  You will have to rely on the context
of its use to determine which it is.
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Dimensional AnalysisDerived Dimensions Notes -
Reduction
SWTJC STEM ENGR 1201
SI (Systeme International)         

1. In every case, derived dimensions are completely
   expressible in terms of fundamental/supplementary
   dimensions as illustrated in the last column. 
   This reflects the reduction process that holds
   for all dimensional systems.

Can m/s be reduced?
No. Has no derived units.
Can N/m2 be reduced?
Yes. N (newton) is a derived unit.
1 In SI, force is derived from mass (F M.L/T2)
making the force unit a kg.m/s2 that is called a
newton (N).2 The use of the letter "A" to refer
to both the dimension and the unit is a bit
confusing.  You will have to rely on the context
of its use to determine which it is.
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Dimensional AnalysisDerived Dimensions Notes -
Coherency
SWTJC STEM ENGR 1201
SI (Systeme International)         

• Another dimensional system characteristic is
  whether or not it is coherent.  A coherent system
  "adheres to the principle that each derived unit
  is a product or quotient of base and
  supplementary units without any conversion
  factors".  

SI and USCS are coherent. AES (American
Engineering System) is not.
From Newtons Second Law, SI and USCS Weight
m . g AES Weight m . g / 32.17
1 In SI, force is derived from mass (F M.L/T2)
making the force unit a kg.m/s2 that is called a
newton (N).2 The use of the letter "A" to refer
to both the dimension and the unit is a bit
confusing.  You will have to rely on the context
of its use to determine which it is.
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Dimensional AnalysisDerived Dimensions Examples
SWTJC STEM ENGR 1201
SI (Systeme International)         

1 In SI, force is derived from mass (F M.L/T2)
making the force unit a kg.m/s2 that is called a
newton (N).2 The use of the letter "A" to refer
to both the dimension and the unit is a bit
confusing.  You will have to rely on the context
of its use to determine which it is.
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Dimensional AnalysisDerived Dimensions Examples
SWTJC STEM ENGR 1201
SI (Systeme International)         

1 In SI, force is derived from mass (F M.L/T2)
making the force unit a kg.m/s2 that is called a
newton (N).2 The use of the letter "A" to refer
to both the dimension and the unit is a bit
confusing.  You will have to rely on the context
of its use to determine which it is.
Note a pascal (Pa) is a newton per square meter
(N/m2)!
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Dimensional AnalysisDerived Dimensions Examples
SWTJC STEM ENGR 1201
SI (Systeme International)         

1 In SI, force is derived from mass (F M.L/T2)
making the force unit a kg.m/s2 that is called a
newton (N).2 The use of the letter "A" to refer
to both the dimension and the unit is a bit
confusing.  You will have to rely on the context
of its use to determine which it is.
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Dimensional AnalysisDerived Dimensions Examples
SWTJC STEM ENGR 1201
SI (Systeme International)         

1 In SI, force is derived from mass (F M.L/T2)
making the force unit a kg.m/s2 that is called a
newton (N).2 The use of the letter "A" to refer
to both the dimension and the unit is a bit
confusing.  You will have to rely on the context
of its use to determine which it is.
DimAnalysis cg13a
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Dimensional AnalysisDerived Dimensions Examples
SWTJC STEM ENGR 1201
SI (Systeme International)         

1 In SI, force is derived from mass (F M.L/T2)
making the force unit a kg.m/s2 that is called a
newton (N).2 The use of the letter "A" to refer
to both the dimension and the unit is a bit
confusing.  You will have to rely on the context
of its use to determine which it is.
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