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```					                                     Dimensional Analysis
SWTJC STEM – ENGR 1201
Definitions

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 Analysis
SWTJC STEM – ENGR 1201
Examples

Dimension   Symbol   Unit        Example

Length      L        meters      Length of room
foot        Stopping distance
Area        L2       meters2     Area of room floor
foot2       Cross-section of I-beam
Time        T        seconds     Stopping time of car
Flight time of arrow
Mass        M        kilograms   Mass of engine block
slug        Mass of electron
Force       F        newtons     Weight of rocket
pounds      Rocket engine thrust
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SWTJC STEM – ENGR 1201                  Dimensional Analysis
Dimension Types

Dimensions are classified as one of three types:

(1) fundamental

(2) supplementary

(3) derived

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Dimensional Analysis
SWTJC STEM – ENGR 1201
Fundamental Dimensions
Fundamental dimensions “Certain fundamental qualities such as length,
mass, force, and time are symbolized with a single letter.”
Fundamental Dimension         Symbol
Length                        L
Mass                          M
Time                          T
Force                         F
Temperature                   q (theta)
Electric current              A
Electric charge               Q
Molecular substance           n
Luminous intensity            I
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Dimensional Analysis
SWTJC STEM – ENGR 1201                    Fundamental Dimensions
Dimensional Systems

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
SWTJC STEM – ENGR 1201                Fundamental Dimensions
SI (Metric)
SI - Systeme International or Metric System

Fundamental Dimension     Symbol              Base Unit
1. Length                 L                   meter (m)
2. Mass                   M                   kilogram (kg)
3. Time                   T                   second (s)
4. Temperature            q (theta)           kelvin (K)
5. Electric current       A                   ampere (A)
6. Molecular substance    n                   mole (mol)
7. Luminous intensity     I                   candela (cd)

Note: Force and charge are not fundamental dimensions.

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SWTJC STEM – ENGR 1201              Dimensional Analysis
Fundamental Dimensions
USCS
USCS - United States Customary System

Fundamental Dimension    Symbol           Base Unit
1. Length                L                foot (ft)
2. Force                 F                pound (lb)
3. Time                  T                second (s)
4. Temperature           t                rankine (R)
5. Electric current      A                ampere (A)
6. Molecular substance   n                mole (mol)
7. Luminous intensity    I                candela (cd)

Note: Mass and charge are not fundamental dimensions.

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Dimensional Analysis
SWTJC STEM – ENGR 1201                  Fundamental Dimensions
Absolute vs. Gravitational
(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!

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Dimensional Analysis
SWTJC STEM – ENGR 1201                 Types of Dimensions
Supplementary

Dimensions are classified as one of three types:

(1) fundamental

(2) supplementary

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SWTJC STEM – ENGR 1201              Dimensional Analysis
Supplementary Dimensions

Supplementary dimensions “Embody geometric concepts needed
in the mathematical formulation of natural laws".

Supplementary Dimensions    Symbol    Unit (abbrev.)

Plane angle                 θ         radian (rad)

Solid angle                 β         steradian (sr)

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Dimensional Analysis
SWTJC STEM – ENGR 1201                 Types of Dimensions
Derived

Dimensions are classified as one of three types:

(1) fundamental

(2) supplementary

(3) derived

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Dimensional Analysis
SWTJC STEM – ENGR 1201                 Types of Dimensions
Derived
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)

Dimension
L                          Area
L       L . L = L2       Symbol
Length                       4 m2
2m
Base Unit
L

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Dimensional Analysis
SWTJC STEM – ENGR 1201                  Types of Dimensions
Derived
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)

F (lb)            Pressure
Force                  (Distributed)       Force /Area
F                                           F/L2
lb (pounds)                                      lb/ft2

L (ft)
L (ft)
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SWTJC STEM – ENGR 1201   Dimensional Analysis
Derived Dimensions
SI

2        2
2       2

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SWTJC STEM – ENGR 1201   Dimensional Analysis
Derived Dimensions
USCS

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Dimensional Analysis
SWTJC STEM – ENGR 1201                  Derived Dimensions
Notes - Reduction

(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.

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Dimensional Analysis
SWTJC STEM – ENGR 1201                   Derived Dimensions
Notes - Coherency

(2) 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 Newton’s Second Law,
SI and USCS            Weight = m . g
AES                    Weight = m . g / 32.17

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Dimensional Analysis
SWTJC STEM – ENGR 1201                         Derived Dimensions
Examples
1. Write newtons in base units. From
Newton' s Second Law,
F  mass(kg)  accelerati (m / s 2 )
on
m           kg  m
N  kg             
2
s            s2

2. Reduce pascals to base units.
m               m
kg              kg 
Pa 
N
            s2             s2
m2            m2            m 21
kg
Pa 
m  s2
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Dimensional Analysis
SWTJC STEM – ENGR 1201                 Derived Dimensions
Examples

3. A forceof 20 N acts on a square plate 2 meters
on a side. Find the pressurein pascals. Since
SI is coherent and P  F / A.
Area of plate  2m  2m  4m 2
20 N      N
P      2
 5 2  5 Pa
4m        m

Note: a pascal (Pa) is a newton per square meter (N/m2)!

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Dimensional Analysis
SWTJC STEM – ENGR 1201                    Derived Dimensions
Examples
4. A pressureof 26 psi (poundsper square inch)
acts on a square plate 1.5 ft on a side. Find the
total force. Since USCS is coherent,we must
work within the base units. Inch is not a base unit,
so inch units must be convertedto feet before
applying the relationF  P  A.

lb       in 2          lb          in 2          lb
P  26        (12 )  26             144        3,744
in 2      ft           in 2         ft 2         ft 2
lb
F  P  A  3,744         (1.5ft  1.5ft )
2
ft
F  8,424 lb or 8.42  103 lb
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Dimensional Analysis
SWTJC STEM – ENGR 1201                       Derived Dimensions
Examples

5. Torricelli ' s Principle
h         Given :
d
A  area of opening [L2 ]
d  diameter of circular opening [L]
h  height of liquid in container [L]
L
g  local accelerati on due to gravity [       2
]
T
L3
Q  heat flow rate [ ]
T
Find :
(a) Verify dimensions of relation for Q is consisten.
(b) Assume the following values and calculate Q.
d  2 cm
h  1.2 m
m
g  9.81
s2
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Dimensional Analysis
SWTJC STEM – ENGR 1201                            Derived Dimensions
Examples

Re altions : FromTorricelli ' s Principle,
h
d   Q  A 2gh Q  A 2gh
Solution :
(a) Verify dimensions of Q from the formula
L               L2 L L3
Q  A 2gh  L        2
L L      2
L      2
, same as Q.
T2            T2       T    s
(b) Calculatin g the area A of a circular opening,
d  2 cm  0.02 m
  d 2   0.02 2
A                     3.14  10  4 m 2
4         4
Q  A 2gh
m
Q  3.14  10  4 m 2 2  9.81          1.2m
s2
3   m3
Q  1.52  10            Ans
s

DimAnalysis cg13a

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