1 Physical Quantity and Physical Relation Functional Form q = f (x 1, x 2, …) There are Two Ways to Determine The Numerical Value of A Physical Quantity

1

Physical Quantity and Physical Relation

Functional Form q = f (x1, x2, …)

There are Two Ways to Determine The Numerical Value of A Physical

Quantity q

Direct Measurement of q Measured Quantity

Determination of q from A Physical Relation for q Derived Quantity

Many Different Physical Principles (for an experiment)

Measured Quantity VS Derived Quantity

Data Reduction Diagram (DRD) for A Physical Quantity q, DRD-q

2145-391 Aerospace Engineering Laboratory I

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Defining An Experiment with

Design of An Experiment Using DRDs

);;( cpxfy Dependent Variable

Independent Variables

Variable Parameters

Constant Parameters


Parameters

C

x (unit x)

y (unit y) p = p1

p = p2

p = p3

3

Physical Quantity and Physical Relation

Functional Form ),...,( 21 xxq

4

Physical QuantityDescribing A Physical Quantity

In an experiment, we want to determine the numerical values of various physical

quantities.

Physical quantity

A quantifiable/measurable attribute we assign to a particular characteristic of

nature that we observe.

)(][,2.1.3.2

LengthLlmlDimensionQmeasureofunitQunitwrtvaluenumerical

][q

Q

Q

Describing a physical quantity q

1. Dimension

2. Numerical value with respect to the unit of measure

3. Unit of measure

5

Physical Relations/Principles

Physical Relation

A relation among physical quantities.

A (valid) physical relation obeys the principle of dimensional homogeneity.

There are many types of physical relations:

Definition [Equality is by definition, := ]

Physical laws/relations [Equality is by law/theory, = ]

Geometric relation (L): sine law, etc.

Kinematic relation (Lt):

Dynamic relation (MLt):

V

MVM :),(

amamF

),(

etc.,,dy

duamF yx

etc.,2

1 2atuts

6

Physical Relation

2

2

1),,( atutatufs

Physical relation: ,...),( 21 xxfq

In a physical relation

q is a function of x1 , x2 , …

q depends on x1 , x2 , …

,...),( 21 xxfq

In order to determine the numerical value of q

1. the physical relation f must be known, and

2. the numerical values of all variables and constants x1 , x2 , … must be known

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Functional Form

Physical relation: ,...),( 21 xxfq

q is a function of x1 , x2 , …

q depends on x1 , x2 , …

the numerical value of q is found from

1. known f , and

2. known values of all x1 , x2 , …

We use parenthesized list of independent variables x1 , x2 , … after q

to indicate that

Functional

form

),...,( 21 xxq

‘sources’ of the numerical value of q

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There are Two Ways to Determine The Numerical Value of A Physical Quantity q

Direct Measurement of q Measured Quantity

Determination of q from A Physical Relation for q Derived Quantity

Many Different Physical Principles (for any one experiment)

Some are based on Measurement and Measured Quantity

Some are based on Physical Relation and Derived Quantity

Example 1

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Example 1: Free Falling ExperimentDetermine the distance the ball travels to ground

S = ?

Class Discussion

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Principle 1: Measurement and Measured Quantity

We can measure s with a measuring tape

Instrument: Measuring tape

The numerical value of s is determined by

measurement with a measuring instrument

s is a measured quantity

(in the current experiment)

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Principle 2: Physical Relation and Derived Quantity

Instrument: 1) stop watch (t)

t : stop watch

g: look up in a reference

We can calculated/derived the numerical value of s from

2

2

1),( gttgs

The numerical value of S is determined from

1. the known physical relation f : , and

2. the known numerical values of all other variables (t and

g) in the relation f

s is a derived quantity

(in the current experiment)

2

2

1),( gttgs

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The Determination of The Numerical Value of A Physical Quantity qMeasured Quantity VS Derived Quantity

The Determination of The Numerical Value of A Physical Quantity q

In a physical phenomenon / current experiment, the numerical value of a physical quantity q can be

(and must be) determined either through

measurement with a measuring instrument Measured Quantity

or

derived through a physical relation Derived Quantity

s is a measured quantity

Its numerical value is determined via

measurement with a measuring

instrument.

s: measuring tape

Instrument: 1) measuring tape (s)

s is a derived quantity

Its numerical value is determined through

1. known physical relation f, and

2. known numerical values of all other variables in f

Instrument: 1) stop watch (t)

2

2

1),( gttgs

t : stop watch

g: look up in a reference

Physical relation

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must be either through

Measurement with an instrument Measured quantity

or

Derived through a physical relation Derived quantity(and by no other means)

Because of existing physical relations/laws, we don’t want anybody to make up any

number for a physical quantity.

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Many Different Physical Principles (for an experiment)

Additional Example

Example 2

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Example 2: Experiment – Determine the density of gas

Experiment: Determine the density of gas in a closed container.

Class Discussion(on the principles that we can use to conduct an experiment)

How can we find out the density of gas in this

closed container?

Is there just one way or are there many ways?

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Physical Principles for An Experiment

Experiment: Determine the density of gas in a closed container.

There can be many physical principles (more than one) that we can use to conduct an

experiment and determine the numerical value of the desired physical quantity q.

V

MVM :),(

Principle 2: Use the perfect gas law

(Thermodynamic definition/relation for density under

specialized condition)

Instruments:

1. Pressure gage to measure pressure (p) in the unit of

pressure, pa

2. Thermometer to measure temperature (T ) in the unit of

temperature oC

Need to know gas to determine the gas constant R.

RT

pTRp ),,(

Pressure gage (p)

Thermometer (T)

Principle 1: Use mechanical definition of density

Instruments:

1. Scale to measure masses (M ) in the unit of

mass, kg

2. Measuring tape to determine volume (V)

V

MVM :),(

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(in a current experiment)

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must be either through

Measurement with an instrument Measured quantity

or

Derived through a physical relation Derived quantity(and by no other means)

Because of existing physical relations/laws, we don’t want anybody to make up any

number for a physical quantity.

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Measured QuantityIs it a measured quantity or a derived quantity? (in the current experiment)

A measured quantity q

is the quantity whose numerical value is read from the instrument in the

unit of q directly.


Instruments:

1. Scale to measure masses (M ) in the unit of

mass, kg


V

MVM :),(

M is a measured quantity

its numerical value is read from the instrument

(scale) in the unit of mass (kg) directly

is a derived quantity

its numerical value is derived from the physical

relation = M/V.

V ?

, M, V: Are they measured or derived?

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, M, V : Are they measured or derived?

M is a measured quantity

its numerical value is read from the instrument (scale) in the unit of mass (kg) directly

}S/N....Scale,:{ kgM q { measured unit: Measuring instrument identity }

source of the numerical value of q is in braces.

is a derived quantity

its numerical value is derived from

1) the physical relation = M/V, and

2) known values of M and V.

]/[:),( 3mkgV

MVM

}S/N....Scale,:{ kgM ][.... 3mV

]t[,...),(,..., 2121 )( uniderived relationexplicit

righttheonconstantsandvariablesalloflist

xxfxxq

source of the numerical value of q is in parentheses.

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V ? Do we really measure volume using an instrument from which the

numerical value of volume is read directly in the unit of volume (e.g., m3)?

Method 1: Use mechanical definition of density

Instruments:

1. Scale to measure masses (M ) in

the unit of mass, kg


V

MVM :),(

}...#tapeMeasuring:{mh}....#tapeMeasuring:{md

][4

),( 32 mhdhdV

V is a derived quantity

its numerical value is derived from

1) the physical relation , and

2) known values of measured quantities d and h.

hdhdV 2

4),(

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Measured QuantityIs it a measured quantity or a derived quantity, really? (in the current experiment)

Look at the unit of the instrument!

If you don’t read its unit from the measuring instrument, it

is not a measured quantity.

A measured quantity q

is the quantity whose numerical value is read from the instrument in the

unit of q directly.

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In Summary: Measured Quantity VS Derived Quantity

Measured Quantity q: The numerical value of a measured quantity is

determined directly by measurement with a measuring

instrument, which reads out in the unit of q directly.

Derived quantity q :

The numerical value of a derived quantity is

determined

1. through a known physical relation f,

and 2. known values of all variables and constants

(Without knowing both 1 and 2 completely, we cannot find the numerical value of a derived

quantity q.)

,...),( 21 xxfq

,...),( 21 xxf

,..., 21 xx

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Data Reduction Diagram (DRD)

for A Physical Quantity q, DRD-q

(for any one physical quantity in an experiment)

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KEY IDEA for A DRD-q

A diagram that we can trace clearly, specifically, and systematically

1. the sources of the numerical values that enter our experiment at the source level

[source-level / bottommost-level boxes],

and

2. the transformations of numerical values [derived-box / data-analysis boxes]

from the source-level values,

through various physical / derived relations in the current experiment,

to the final value of the desired variable q.

A DRD for A Physical Quantity q

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Data Reduction Diagram (DRD)


Instruments:

1. Scale to measure masses (M )

2. Measuring tape to determine

volume (V)

V

MVM :),(

Experiment: Determine the density r of gas in a closed container.

Bottommost level = Braced Boxes / Measured

quantities only

]/[:),( 3mkgV

MVM

}...S/NScale,:{ kgM }#...tapeMeasuring:{mh

}#...tapeMeasuring:{md

][4

),( 32 mhdhdV

DRD -

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Example 3: DRD

Class Discussion

Construct a DRD for (the determination

of the numerical value of) the density

V

MVM :),(

Principle 2: Use the perfect gas law

(Thermodynamic definition/relation for density under

specialized condition)

Instruments:


pressure, pa

2. Thermocouple to measure temperature (T ) in the unit of

temperature oC


RT

pTRp ),,(

Pressure gage (p)

Thermocouple (T)

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Instruments:


pressure, pa

2. Thermometer to measure temperature (T ) in the unit of

temperature oC


]/[),,( 3mkgRT

pRTp

Pressure gage (p)

Thermocouple (T)

???R}S/N...gage,Pressure:{pap }S/N...r,Thermomete:{ CT o

][15.273)())(( KCTCTT oo Unit conversion

Because there is a transformation of a numerical value through a relation, we consider

unit conversion as one of the data analysis step.

This is a derived box (parenthesized box).

What kind of quantity is R, measured or derived?

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Referenced Quantity

For some quantities, we may not be able to measure or derive it directly in

the current experiment.

We take their numerical value from some reference source.

We refer to this kind of quantities in the current experiment as

Reference Quantities

Regardless, being a physical quantity, the numerical value of a reference

quantity must be either

measured, or

derived

by the original author of the value.

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Derived-Referenced Quantity VS Stated-Referenced Quantity

Derived-Referenced Quantity

Example: Determination of density from

1. thermodynamic table, and

2. known values of p and T (and type of gas,

tg)

2007,...}Boles,andCengel4,-ATable

:{kg/mTablemicThermodyna)tg,,( 3Tp

}S/N...gage,pressure:{ pap


}S/N...r,Thermomete:{ CT o

)gasoftypetg(

}......{

tg

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Although the physical relation is not stated explicitly as an equation,

table,

chart,

etc.,

have an underlying physical relation.

We need to know the numerical values of p and T first before we can look up the

table to get .

The numerical value of depends on the numerical values of p and T.

{.....}TablemicThermodyna)tg,,( Tp

Use functional form and parentheses for a derived quantity.

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Derived-Referenced Quantity VS Stated-Referenced Quantity

Stated-Referenced Quantity

Example:

710} p. D/2, Table York, New Wiley,Edition,Fourth

Dynamics :Mechanics gEngineerin 1998, Kraig,andMeriam:/{ 2 smg

In this case, g is not a derived-referenced quantity.

Its numerical value is looked up directly, without the knowledge of the numerical

values of other quantities.

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In this case, g is a derived-referenced quantity since we take that it depends on

the elevation h.

710. p. D/2, Table York, New Wiley,Edition,Fourth

]2

[m/sDynamics :Mechanics gEngineerin 1998, Kraig,andMeriam)( hg

....}:{ mh

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Back to Example 3 Pressure gage (p)

Thermocouple (T)

]/[),,( 3mkgRT

pRTp

}S/N...gage,Pressure:{pap }S/N...r,Thermomete:{ CT o


p.910}1,-ATable,2007,.....Boles,andCengel:/{ KkgJR

DRD -

Source / Bottommost Level - Braced Boxes only

This is where numerical values first enter our experiment

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Summary of Types and Boxes of Quantities in DRDConvention for Boxes of Various Types of Quantities in DRD

1. Measured Quantity q

q { measured unit: Measuring instrument identity }

Measured unit is the unit that is read directly from the instrument, no unit conversion.

}#4gagepressureLab:{ pap

[Braced box, source-level box.

Use braces on the LHS.]

source of the numerical value of q

2. Derived Quantity q

]t[,...),(),...,( 2121 uniderived relationexplicit

righttheonconstantsandvariablesalloflist

xxfxxq

]p[),,( aRTRTp

Derived unit is the unit that is a result of the physical relation f and the actual units that correspond to the

numerical values of x1, x2, … that are input into the relation f, no unit conversion.

[Parenthesized-box, derived box.

Use parentheses on the LHS]


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3.1. Derived-Referenced Quantity q

Reference}:{tablemicThermodyna),...,( 21 unitreference

relationderivedtheininvolvedconstants

andvariablesalloflist

xxq

916}p.2007,...,Boles,andCengel4,-ATable

:{kg/mTablemicThermodyna)tg,,( 3Tp

[Parenthesized-box, derived box.

Use parentheses on the LHS]


Reference unit is the unit that corresponds to the numerical value that is given in the reference, no unit

conversion.

3.2. Stated-Referenced Quantity q

Reference}:{ unitreferenceq

710} p. D/2, Table York, New Wiley,Edition,Fourth

Dynamics :Mechanics gEngineerin 1998, Kraig,andMeriam:/{ 2 smg

[Braced box, source-level box.

Use braces on the LHS.]


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Summary of Rules and Guides for a DRD

1. Braced-Boxes / Source Level

At the bottommost/source level only, and nowhere else.

2. Parenthesized-Boxes / Derived Levels

Can never be at the bottommost/source level since they need

sources of numerical values from somewhere else.

q { …. }

q ( …. )

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3. Numerical Transformation

Every step of numerical transformation from the

bottommost/source/braced-box level to the DRD-variable (q) must be recorded in

the DRD [via a derived/parenthesized box].

Relations that result in corresponding numerical transformations are

definition,

physical relation (geometrical, kinematical, and dynamical relation),

calibration relation,

unit conversion,

etc.

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4. Unit

Every box in a DRD must have the corresponding unit stated.

Various types of units (terminology by convention)

Measured unit

Derived unit

Reference unit

A derived unit in a derived box in a DRD must be consistent with both

the units of the source variables of that box, and

the relation in that box.

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Workshop for A DRD for A Single Quantity q

41

Defining An Experiment with

);;( cpxfy Dependent Variable


Variable Parameters

Constant Parameters


Parameters

C

x (unit x)

y (unit y) p = p1

p = p2

p = p3

42

Defining An Experiment

Often in an experiment, the objective is not simply to find a single value

of a single physical quantity but

QUESTION:

‘whether and how y is related to x under the condition ( p , c ):

a physical relation:

);;( cpxfy

43

C

x (unit x)

y (unit y) p = p1

p = p2

p = p3

Experiment: );;( cpxfy

Dependent Variable


Variable Parameters

Constant Parameters


Parameters

44

Example

T (K)

p (pa) = 1

= 2

= 3

Fixed gas (R)

Isochoric process

p is dependent variable

T is independent variable

is variable parameter

R (tg) is constant parameter

QUESTION:

‘whether and how the pressure p is related to the

temperature T under the condition of various

density and constant gas type (R).

Physical relation: ))tg(;;( RTfp

);;( cpxfy

x (unit x)

y (unit y)p = p1

c

Line of constant p

p = p2

p = p3

45

Example

From Abbot, I. R. H. and von Doenhoff, A. E., 1959, Theory of Wing Sections: Including A Summary of Airfoil Data, Dover, pp. 496-497.

y = cl

x = (deg)

p = Re

p1

p1

46

Design of An Experiment

Using DRD and Its Consequences

47

Design of An Experiment Using DRD and Its Consequences

1. Question/Relation

Set the goal that we want to answer the question

‘whether and how y is related to x under the condition ( p , c ):

Experiment: y = f ( x ; p ; c )

2. Graphical Representation of Results

We then know that the graphical representation of the relation

should look like this:

C

x (unit x)

y (unit y) p = p1

p = p2

p = p3

y = f ( x ; p ; c )

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3. Data Reduction Diagram (DRD)

Construct a data reduction diagram (DRD) for each of the final

variables: y, x, p, and c

DRD-y

DRD-x

DRD-p

DRD-c

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4. From this set of DRDs for the whole experiment

1. All The Measured Quantities and Instruments

Measured Quantities: We know all of the measured quantities in this

experiment from the bottommost/source level

Instruments: We know all of the instruments we need

for this experiment

DCW: We can construct a data-collection

worksheet.

All The Derived Quantities and Physical Relations

Derived Quantities and Physical Relations: We know all of the

derived quantities and all of the corresponding physical relations.

DAW: We can construct a data-analysis worksheet.

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3. All The Referenced Quantities and Their Sources

4. Diagnostic Tool

We can use the set of the DRDs to check our experiment when we

expect that something might have gone wrong.

5. Uncertainty Analysis

Later on, we will also use this set of DRDs as a guide for uncertainty

analysis.

Documents

1 Physical Quantity and Physical Relation Functional Form q = f (x 1, x 2, …) There are Two Ways to Determine The Numerical Value of A Physical Quantity