dimension “a physical property being measured". swtjc stem – engr 1201 dimanalysis cg13a...

Dimension “A physical property being measured".

SWTJC STEM – ENGR 1201

DimAnalysis cg13a

Dimensional AnalysisDefinitions

Qualitative – Asks what?

Quantitative – How much?

Unit “Addresses the quantitative aspect of a dimension".


DimAnalysis cg13a

Dimensional AnalysisExamples

Dimension Symbol Unit Example

Length L metersfoot

Length of room

Stopping distance

Area L2 meters2

foot2

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

Dimensions are classified as one of three types:

(1) fundamental

(2) supplementary

(3) derived


DimAnalysis cg13d

Dimensional AnalysisDimension Types

Fundamental Dimension Symbol

Length L

Mass M

Time T

Force F

Temperature (theta)

Electric current A

Electric charge Q

Molecular substance n

Luminous intensity I


DimAnalysis cg13a

Dimensional AnalysisFundamental Dimensions

Fundamental dimensions “Certain fundamental qualities such as length, mass, force, and time are symbolized with a single letter.”

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)


DimAnalysis cg13a

Dimensional Analysis Fundamental Dimensions

Dimensional Systems


DimAnalysis cg13a


SI (Metric)

SI - Systeme International or Metric System


1. Length L

2. Mass M

3. Time T

4. Temperature (theta)

5. Electric current A

6. Molecular substance n

7. Luminous intensity I

Base Unit

meter (m)

kilogram (kg)

second (s)

kelvin (K)

ampere (A)

mole (mol)

candela (cd)

Note: Force and charge are not fundamental dimensions.

http://physics.nist.gov/cuu/Units/


DimAnalysis cg13a


USCSUSCS - United States Customary System


1. Length L

2. Force F

3. Time T

4. Temperature t

5. Electric current A

6. Molecular substance n

7. Luminous intensity I

Base Unit

foot (ft)

pound (lb)

second (s)

rankine (R)

ampere (A)

mole (mol)

candela (cd)

Note: Mass and charge are not fundamental dimensions.



DimAnalysis cg13a

Dimensional Analysis 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!



(1) fundamental


DimAnalysis cg13a

Dimensional AnalysisTypes of Dimensions

Supplementary

(2) supplementary


DimAnalysis cg13a

Dimensional Analysis Supplementary Dimensions

Supplementary Dimensions Symbol Unit (abbrev.)

Plane angle θ radian (rad)

Solid angle β steradian (sr)

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



(1) fundamental

(2) supplementary


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Derived

(3) derived

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


DimAnalysis cg13a


Derived

Derived dimensions “Associated with physical properties that can be written as some combination of fundamental dimensions".

L

L

LLength

2 m

AreaL . L = L2

4 m2

Base Unit

Symbol

Dimension

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)


DimAnalysis cg13a


Derived

Derived dimensions “Associated with physical properties that can be written as some combination of fundamental dimensions".

ForceF

lb (pounds)

L (ft)

L (ft)

F (lb)(Distributed)

PressureForce /Area

F/L2

lb/ft2


DimAnalysis cg13a

Dimensional Analysis Derived Dimensions

SI

2 2

2 2



DimAnalysis cg13a

Dimensional Analysis Derived Dimensions

USCS



DimAnalysis cg13a

Dimensional AnalysisDerived 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.


DimAnalysis cg13a


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


DimAnalysis cg13a


Examples

2

22

22

2

sm

kgPa

m

s

mkg

m

s

mkg

m

NPa

units. base to pascals Reduce 2.

s

mkg

s

mkgN

)s /m(onaccelerati)kg(massF

Law, Second sNewton'

From units. base in newtons Write1.

1222


DimAnalysis cg13a


Examples

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

Pa 554

20

422plate of Area

A. / F P andcoherent is SI

Since pascals.in pressure theFind side. aon

meters 2 plate square aon acts N 20 of forceA 3.

22

2

m

N

m

NP

mmm


DimAnalysis cg13a


Examples

lb . or lb ,F

)ft.ft.(ft

lb,APF

ft

lb,

ft

in

ni

lb)

ft

in(

in

lbP

A.PF relation the applying

before feet to converted be must units inch so

unit, base a not is Inch units. base the withinwork

must wecoherent, is USCS Since force. total

the Find side. a on ft 1.5 plate square a on acts

inch) square per (pounds psi 26 of pressure A 4.

3

2

22

2

22

2

104284248

51517443

7443144261226


DimAnalysis cg13a


Examples

h

d

2

3

2

2

s

m 9.81g

m 1.2h

cm 2d

Q. calculate and valuesfollowing theAssume (b)

consisten. is Qfor relation of dimensionsVerify (a)

:Find

]T

L[ rate flow heatQ

]T

L[gravity todueon accelerati local g

[L]container in liquid ofheight h

[L] openingcircular ofdiameter d

]L[ opening of area A

:Given

Principle s'Torricelli 5.


DimAnalysis cg13a


Examples

h

d

Ans s

m 1052.1Q

m2.1s

m81.92m1014.3Q

gh2AQ

m1014.34

02.0

4

d A

m 0.02 cm 2 d

opening,circular a ofA area thegCalculatin (b)

Q. as same , s

L

T

LL

T

LLL

T

LLgh2A Q

formula thefrom Q of dimensionsVerify (a)

:Solution

gh2A Qgh2A Q

Principle, s'TorricelliFrom:altionsRe

33

224

2422

32

2

22

22

dimension “a physical property being measured". swtjc stem – engr 1201 dimanalysis cg13a...

Documents

fundamental swtjc stem

uscs system

derivedswtjc stem engr

gravitational system

absolute system

force unit

metric system note

definition of force