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Unit Analysis

SWTJC STEM – ENGR 1201. Unit Analysis. Unit Analysis “Measurement units can be manipulated in a similar way to variables in algebraic relations.”. The basis for this analysis is embodied in three rules: Dimensional Consistency Rule Algebraic Manipulation Rule Transcendental Function Rule.

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Unit Analysis

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  1. SWTJC STEM – ENGR 1201 Unit Analysis Unit Analysis “Measurement units can be manipulated in a similar way to variables in algebraic relations.” • The basis for this analysis is embodied in three rules: • Dimensional Consistency Rule • Algebraic Manipulation Rule • Transcendental Function Rule Content Goal 15

  2. SWTJC STEM – ENGR 1201 Rule #1 – Dimensional Consistency Rule In a unit relation, all terms must be dimensionally consistent.  This means that each term must have the same units or be reducible to the same units. Rule 1 Dimensional Consistency Dimension refers to “what is being measured”. For instance, when measuring the length of a table, “length” is the dimension. The unit could be feet, meters, or a variety of other “length” units. Reducible refers to rewriting the units in fundamental units. Units are either fundamental or derived. Derived units are combinations of eight fundamental units. Refer to Derived Units Charts on “Useful Links” Content Goal 15

  3. SWTJC STEM – ENGR 1201 Adding Apples and Oranges TheDimensional Consistency Rulesimply reinforces the common sense idea that you can only add and subtract identical things. You cannot mix apples and oranges! ? Content Goal 15

  4. SWTJC STEM – ENGR 1201 Applying Consistency Rule Example x = a + b - c (Terms are separated by addition or subtraction) terms (length) = (length) + (length) – (length) (meters) = (meters) + (meters) – (meters) Consistencymeans terms “x”, “a”, “b”, and “c” must have the same dimension, i.e. units, in this case length/meters. If “x” is length (meters), all other terms must be length (meters)! Note that (meters) - (meters) = (meters) not zero! 16 meters - 12 meters = 4 meters! Content Goal 15

  5. SWTJC STEM – ENGR 1201 Rule 2 Algebraic Manipulation Rule #2 – Algebraic Manipulation Rule Unit relations that are multiplied and/or divided can be treated like variables; i.e., canceled, raised to powers, etc. During algebraic manipulation of a relation, dimensional consistency must be maintained.  When finished, if dimensional inconsistency is noted, then either an algebraic manipulation error occurred or the original formulation of the relation was faulty. Working through the units is a great way to check your algebra! Content Goal 15

  6. Distance formula: d = v0. t + (1/2) . a . t where d (m), v0 (m/s), t (s), and a (m/s2) m = (m/s) . s + (none) . (m/s2) . sm = m + m/sA problem? The relation isinconsistent! Formula is incorrect! Distanceformula: d = v0. t + (1/2) . a . t2m = (m/s) . s + (none) . (m/s2) . s2m = m + mThe formula isconsistent! SWTJC STEM – ENGR 1201 dimensionless constant (no units!) Distance Formula Example Examples Content Goal 15

  7. SWTJC STEM – ENGR 1201 Particle Energy Example DimAnalysis cg13d

  8. SWTJC STEM – ENGR 1201 Particle Energy Example DimAnalysis cg13d

  9. SWTJC STEM – ENGR 1201 Particle Energy Example DimAnalysis cg13d

  10. SWTJC STEM – ENGR 1201 Rule 3 Transcendental Function Rule #3 – Transcendental Function Rule Transcendental functions (trig, exponential, etc.) and their arguments cannot have dimensions (units). Examples of transcendental functions includes: sin(x), cos(x), tan(x), arcsin(x) ex log(x), ln(x) Content Goal 15

  11. SWTJC STEM – ENGR 1201 Transcendental Function Examples Consider the relation A = sin(a t + b). Neither (a t + b) nor A can have a unit.  Note that a and t can have units provided they cancel.  Variable b cannot! Suppose a = 6 Hz, t = 10 s, and b = 5 (no units). Hz is the derived unit Hertz and is reducible to fundamental units 1/s. Then A = sin(6 1/s  10 s + 5) = sin(60 + 5) = sin(65) = 0.906 No units! Content Goal 15

  12. SWTJC STEM – ENGR 1201 Richter Scale Example Earthquakeintensity is measured on the Richter Scale. MR= log(A) where is a seismic amplitude factor. The famous San Francisco earthquake of 1906 was MR = 7.8 on the Richter scale. Does A have units? No! According to Rule 3, Transcendental Function DoesMRhave units? No! Ditto. Content Goal 15

  13. SWTJC STEM – ENGR 1201 Bernoulli Example Content Goal 15

  14. SWTJC STEM – ENGR 1201 Bernoulli Example Content Goal 15

  15. SWTJC STEM – ENGR 1201 Coherent Systems of Units A system of units is coherent if all units use the numerical factor of one. For example, the SI system is coherent, so the unit m/s implies (1 meter) / (1 second). Both the SI and USCS systems are coherent. This means that when you use SI or USCS units in a relation (formula), no numeric factors will be needed. Unless otherwise indicated, change all units to SI and USCS base or derived units before plugging in a formula. Content Goal 15

  16. FundamentalDimension Base Unit 1. Length meter (m) 2. Mass kilogram (kg) 3. Time second (s) 4. Temperature kelvin (K) 5. Electric current ampere (A) 6. Molecular substance mole (mol) 7. Luminous intensity candela (cd) SWTJC STEM – ENGR 1201 Base Units SI (Metric) SI - Systeme International or Metric System Note: Force and charge are not fundamental units. DimAnalysis cg13a

  17. SWTJC STEM – ENGR 1201 Derived Units in SI Content Goal 15

  18. Fundamental Dimension Base Unit 1. Length foot (ft) 2. Force pound (lb) 3. Time second (s) 4. Temperature rankine (R) 5. Electric current ampere (A) 6. Molecular substance mole (mol) 7. Luminous intensity candela (cd) SWTJC STEM – ENGR 1201 Base Units USCS USCS - United States Customary System Note: Mass and charge are not fundamental dimensions. DimAnalysis cg13a

  19. SWTJC STEM – ENGR 1201 Derived Units in USCS Content Goal 15

  20. SWTJC STEM – ENGR 1201 Coherent System Example What is the kinetic energy of a 20 ton ship moving 5 knots? What is the system of units? Related to USCS: 1 ton = 2000 lbs, 1 knot = 1.688 ft/s Related to USCS: 1 ton = 2000 lbs, 1 knot = 1.688 ft/s Is USCS coherent? Yes. What is the relation for kinetic energy? Ke = W · v2 / (2· g) where Ke (lb·ft), W (lb), v (ft/s), g (32.2 ft/s2) Content Goal 15

  21. SWTJC STEM – ENGR 1201 Coherent System Example Ke = W · v2 / (2· g) where Ke (lb·ft), W (lb), v (ft/s), g (32.2 ft/s2) What is first step? Convert to base units. What’s not in base units? W in tons and v in knots. 20 tons · 2000 lbs/ton = 40,000 lbs = 4· 104 lbs 5 knots · 1.688 (ft/s)/knot = 8.44 ft/s Substituting, Ke = 4· 104 lb· (8.44 ft/s) 2 / (2· 32.2 ft/s2) Content Goal 15

  22. SWTJC STEM – ENGR 1201 1 Coherent System Example Ke = 4· 104 · 71.2336 / 64.4 lb· ft 2 /s2· s2/ft Ke = 4.42 · 104 lb·ft Ans Content Goal 15

  23. h d SWTJC STEM – ENGR 1201 Torricelli Example Content Goal 15

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