1. 2145-391 Aerospace Engineering Laboratory I
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

2. Independent VariablesC
x (unit x)
y (unit y)
p = p1
p = p2
p = p3
Defining An Experiment with
Design of An Experiment Using DRDs
Dependent Variable
Independent Variables
Variable Parameters
Constant Parameters
Parameters

3. Physical Quantity and Physical Relation
Functional Form

4. Physical Quantity Describing 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.
Describing a physical quantity q
Dimension
Numerical value with respect to the unit of measure
Unit of measure

5. Physical Relations/Principles
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):

6. Physical Relation Physical relation: In a physical relation
q is a function of x1 , x2 , …
q depends on x1 , x2 , …
In order to determine the numerical value of q
the physical relation f must be known, and
the numerical values of all variables and constants x1 , x2 , … must be known

7. Functional Form
Physical relation:
We use parenthesized list of independent variables x1 , x2 , … after q
to indicate that
Functional form
‘sources’ of the numerical value of q
q is a function of x1 , x2 , …
q depends on x1 , x2 , …
the numerical value of q is found from
known f , and
known values of all x1 , x2 , …

8. 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

9. Example 1: Free Falling Experiment Determine the distance the ball travels to ground
Class Discussion

10. 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)

11. Principle 2: Physical Relation and Derived Quantity
t : stop watch
g: look up in a reference
Instrument: 1) stop watch (t)
We can calculated/derived the numerical value of s from
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)

12. s is a measured quantity
The Determination of The Numerical Value of A Physical Quantity q Measured 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)
t : stop watch
g: look up in a reference
Physical relation

13. Measured Quantity VS Derived Quantity
The Determination of The Numerical Value of A Physical Quantity q
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.

14. Additional Example Example 2
Many Different Physical Principles (for an experiment)
Additional Example
Example 2

15. Example 2: Experiment – Determine the density of gas
Experiment: Determine the density r of gas in a closed container.
How can we find out the density of gas in this closed container?
Is there just one way or are there many ways?
Class Discussion
(on the principles that we can use to conduct an experiment)

16. Physical Principles for An Experiment
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.
Experiment: Determine the density r of gas in a closed container.
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)
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.
Pressure gage (p)
Thermometer (T)

17. Measured Quantity VS Derived Quantity
(in a current experiment)

18. Measured Quantity VS Derived Quantity
The Determination of The Numerical Value of A Physical Quantity q
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.

19. Measured Quantity Is it a measured quantity or a derived quantity
Measured Quantity Is 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.
r, M, V: Are they measured or derived?
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)
M is a measured quantity
its numerical value is read from the instrument (scale) in the unit of mass (kg) directly
r is a derived quantity
its numerical value is derived from the physical relation r = M/V.
V ?

20. r, 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
q { measured unit: Measuring instrument identity }
source of the numerical value of q is in braces.
r is a derived quantity
its numerical value is derived from
1) the physical relation r = M/V, and
2) known values of M and V.
source of the numerical value of q is in parentheses.

21. 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
2. Measuring tape to determine volume (V)
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.

22. Look at the unit of the instrument!
Measured Quantity Is it a measured quantity or a derived quantity, really? (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.
Look at the unit of the instrument!
If you don’t read its unit from the measuring instrument, it is not a measured quantity.

23. 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.)

24. Data Reduction Diagram (DRD)
for A Physical Quantity q, DRD-q
(for any one physical quantity in an experiment)

25. A DRD for A Physical Quantity q
KEY IDEA for A DRD-q
A diagram that we can trace clearly, specifically, and systematically
the sources of the numerical values that enter our experiment at the source level [source-level / bottommost-level boxes],
and
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.

26. Data Reduction Diagram (DRD)
Experiment: Determine the density r of gas in a closed container.
DRD - r
Principle 1: Use mechanical definition
of density
Instruments:
1. Scale to measure masses (M )
2. Measuring tape to determine volume (V)
Bottommost level = Braced Boxes / Measured quantities only

27. Example 3: DRD Class Discussion
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. Thermocouple to measure temperature (T ) in the unit of temperature oC
Need to know gas to determine the gas constant R.
Pressure gage (p)
Thermocouple (T)
Class Discussion
Construct a DRD for (the determination of the numerical value of) the density r

28. What kind of quantity is R, measured or derived?
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.
Pressure gage (p)
Thermocouple (T)
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?

29. Referenced Quantity measured, or derived
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.

30. Derived-Referenced Quantity VS Stated-Referenced Quantity
Example: Determination of density r from
1. thermodynamic table, and
2. known values of p and T (and type of gas, tg)
Unit conversion

31. Use functional form and parentheses for a derived quantity.
Although the physical relation is not stated explicitly as an equation,
table,
chart,
etc.,
have an underlying physical relation.
Use functional form and parentheses for a derived quantity.
We need to know the numerical values of p and T first before we can look up the table to get r.
The numerical value of r depends on the numerical values of p and T.

32. Derived-Referenced Quantity VS Stated-Referenced Quantity
Example:
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.

33. In this case, g is a derived-referenced quantity since we take that it depends on the elevation h.

34. Back to Example 3 DRD - r Unit conversion
Pressure gage (p)
Thermocouple (T)
DRD - r
Unit conversion
Source / Bottommost Level - Braced Boxes only
This is where numerical values first enter our experiment

35. source of the numerical value of q source of the numerical value of q
Summary of Types and Boxes of Quantities in DRD Convention 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.
[Braced box, source-level box.
Use braces on the LHS.]
source of the numerical value of q
2. Derived Quantity q
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]
source of the numerical value of q

36. source of the numerical value of q source of the numerical value of q
Derived-Referenced Quantity q
[Parenthesized-box, derived box.
Use parentheses on the LHS]
source of the numerical value of q
Reference unit is the unit that corresponds to the numerical value that is given in the reference, no unit conversion.
Stated-Referenced Quantity q
[Braced box, source-level box.
Use braces on the LHS.]
source of the numerical value of q

37. Summary of Rules and Guides for a DRD
Braced-Boxes / Source Level
At the bottommost/source level only, and nowhere else.
Parenthesized-Boxes / Derived Levels
Can never be at the bottommost/source level since they need sources of numerical values from somewhere else.
q { …. }
q ( …. )

38. Summary of Rules and Guides for a DRD
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.

39. Summary of Rules and Guides for a DRD
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.

40. Workshop for A DRD for A Single Quantity q

41. Independent VariablesC
x (unit x)
y (unit y)
p = p1
p = p2
p = p3
Defining An Experiment with
Dependent Variable
Independent Variables
Variable Parameters
Constant Parameters
Parameters

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:

43. Independent Variables
Variable Parameters
Constant Parameters
Parameters
Experiment: C
x (unit x)
y (unit y)
p = p1
p = p2
p = p3

44. Example y (unit y) p = p1 p = p2 p (pa) r = r1 p = p3 r = r2 r = r3 c
x (unit x)
y (unit y)
p = p1 c
Line of constant p
p = p2
p = p3
T (K)
p (pa)
r = r1
r = r2
r = r3
Fixed gas (R)
Isochoric process
QUESTION:
‘whether and how the pressure p is related to the temperature T under the condition of various density r and constant gas type (R).
Physical relation:
p is dependent variable
T is independent variable
r is variable parameter
R (tg) is constant parameter

45. Example y = cl p = Re x = a (deg) p1
From Abbot, I. R. H. and von Doenhoff, A. E., 1959, Theory of Wing Sections: Including A Summary of Airfoil Data, Dover, pp
y = cl
x = a (deg)
p = Re p1

46. Design of An Experiment Using DRD and Its Consequences

47. Design of An Experiment Using DRD and Its Consequences
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 )
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 )

48. 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

49. All The Measured Quantities and Instruments
From this set of DRDs for the whole experiment
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.

50. All The Referenced Quantities and Their Sources
Diagnostic Tool
We can use the set of the DRDs to check our experiment when we expect that something might have gone wrong.
Uncertainty Analysis
Later on, we will also use this set of DRDs as a guide for uncertainty analysis.