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2145-391 Aerospace Engineering Laboratory I

2145-391 Aerospace Engineering Laboratory I. 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 q Direct Measurement of q  Measured Quantity

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2145-391 Aerospace Engineering Laboratory I

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

  3. Physical Quantity and Physical Relation • Functional Form

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

  5. 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): Physical Relations/Principles Physical Relation • A relation among physical quantities. • A (valid) physical relation obeys the principle of dimensional homogeneity.

  6. Physical relation: In a physical relation • qis a function ofx1 , x2 , … • qdepends onx1 , x2 , … • In order to determine the numerical value ofq • the physical relationf must be known, and • the numerical values of all variables and constantsx1 , x2 , … must be known Physical Relation

  7. Physical relation: We use parenthesized list ofindependent variables x1 , x2 , … after q to indicate that Functional form ‘sources’ of the numerical value of q Functional Form • qis a function ofx1 , x2 , … • qdepends onx1 , x2 , … • the numerical value ofqis 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 ofq  Measured Quantity • Determination of qfrom 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. S = ? Example 1: Free Falling ExperimentDetermine 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. We can calculated/derived the numerical value ofsfrom The numerical value ofS 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) t : stop watch g: look up in a reference Principle 2: Physical Relation and Derived Quantity Instrument: 1) stop watch (t)

  12. sis a derived quantity Its numerical value is determined through 1. known physical relationf, and 2. known numerical values of all other variables in f sis a measured quantity Its numerical value is determined via measurement with a measuring instrument. t : stop watch g: look up in a reference Physical relation s: measuring tape Instrument: 1) measuring tape (s) Instrument: 1) stop watch (t) The Determination of The Numerical Value of A Physical Quantity qMeasured Quantity VS Derived Quantity • The Determination of The Numerical Value of A Physical Quantityq • 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

  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. 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 outthe density of gas in this closed container? • Is there just one wayorare there many ways? Class Discussion (on the principles that we can use to conduct an experiment)

  16. 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) 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.

  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. 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) 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. 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 r is a derived quantity its numerical value is derived from the physical relation r = M/V. V ?

  20. M is a measured quantity its numerical value is read from the instrument (scale) in the unit of mass (kg) directly source of the numerical value of q is in braces. q{measuredunit: Measuring instrument identity } 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. r, M, V : Are they measured or derived?

  21. 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. 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)?

  22. Measured QuantityIs 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 qdirectly. 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. Derived quantity q : • The numerical value of a derived quantity is determined • 1. through a known physical relationf, • 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.) 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 ofq directly.

  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. 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 Data Reduction Diagram (DRD) Experiment: Determine the density r of gas in a closed container. DRD - r

  27. 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) Example 3: DRD Class Discussion Construct a DRD for (the determination of the numerical value of) the density r

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

  29. 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.

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

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

  35. [Braced box, source-level box. Use braces on the LHS.] 1. Measured Quantity q q { measuredunit: Measuring instrument identity } source of the numerical value of q Measured unit is the unit that is read directly from the instrument, no unit conversion. [Parenthesized-box, derived box. Use parentheses on the LHS] 2. Derived Quantity q source of the numerical value of 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. Summary of Types and Boxes of Quantities in DRDConvention for Boxes of Various Types of Quantities in DRD

  36. [Parenthesized-box, derived box. Use parentheses on the LHS] 3.2. Stated-Referenced Quantity q source of the numerical value of q 3.1. Derived-Referenced Quantity q [Braced box, source-level box. Use braces 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.

  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 transformationfrom 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 Variables Parameters Dependent Variable Independent Variables p = p1 p = p2 p = p3 y (unit y) Variable Parameters Constant Parameters C x (unit x) • Defining An Experiment with

  42. QUESTION: ‘whether and how y is related to x under the condition ( p , c): a physical relation: Defining An Experiment Often in an experiment, the objective is not simply to find a single value of a single physical quantity but

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

  44. y (unit y) Line of constant p p = p1 p = p2 p (pa) Isochoric process r = r1 p = p3 r = r2 c r = r3 x (unit x) Fixed gas (R) T (K) 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: Example p is dependent variable T is independent variable r is variable parameter R (tg) is constant parameter

  45. y = cl p = Re p1 p1 x = a (deg) 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. Example

  46. Design of An Experiment Using DRD and Its Consequences

  47. p = p1 p = p2 p = p3 y (unit y) C x (unit x) 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: 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. 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.

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