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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

Chabot Mathematics. Units & Conversions. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. Units Introduction. People measure quantities through comparisons with standards. Every measured quantity has an associated “unit” Which is the name of the Standard.

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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

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  1. Chabot Mathematics Units &Conversions Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. Units Introduction • People measure quantities through comparisons with standards. • Every measured quantity has an associated “unit” Which is the name of the Standard. • Need to define sensible and practical "units" and "standards" that scientists & engineers everywhere can agree upon • Even though there exist an almost infinite number of different physical quantities, we need no more than a handful of “base” standards.

  3. SI System of Units • Système International d'Unités (International System of Units) • A CompletelyConsistentSet of BasicUnits • Requires NO Conversion factors • e.g., 5280 ft = 1 mile • Defined by UNCHANGING Physical Phenomena • Except for one... http://www.bipm.org/en/si/

  4. SI System History • In 1960 The 11th General Conference on Weights and Measures (GCWM) adopted the name SI System, for the recommended practical system of units of measurement. • The 1960 GCWM Specified Seven well-defined “Base” units which, by convention, are regarded as DIMENSIONALLY INDEPENDENT http://www.bipm.org/en/si/

  5. ¿¿ Reader Question ?? • Have you Heard of the unit: • METER (m) • KILOGRAM (kg) • SECOND (s) • AMPERE or AMP (A) • KELVIN (K) • MOLE (mol) • CANDELA (cd

  6. SI Base Units • From this List Observe • Very common Units • Mass (kg) • Length (m) • Time (s) • Some Not so Common Units • Current (A) • Temperature (K) • Some Uncommon Units • Substance amt (mol) • Luminous Int (cd) • All but the kg are defined by Physical Phenomena • Examine the Defs

  7. Length or Distance (meter) 1 meter Laser 1/299792458 s photon Meter Defined • “The path traveled by light in vacuum during a time interval of 1/299792458 of a second.”

  8. Mass (kilogram) kilogram Defined • a cylinder of PLATINUM-IRIDIUM alloy maintained under vacuum conditions by the International Bureau of Weights and Measures in Paris If The ProtoType Were Cubic, its Edge Length would be About 36.2 mm (1.42”); quite small

  9. Time (Second) Second Defined • The duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom • This is the Definition of an “Atomic” Clock • more than 200 atomic clocks are located in metrology institutes and observatories in more than 30 countries around the world

  10. Electric Current (ampere) Amp Defined • That constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 m apart in a vacuum, would produce between these conductors a force equal to 2 x 10−7Newton per metre of length. • What’s a Newton?→ 1kg-m/(s2)

  11. Thermo-dynamic temperature (Kelvin) Kelvin (Temperature) Defined • The unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. • 273.16K = 0.0098 °C • Room Temperature (72 °F) is about 295.5 Kelvins • NO “Degree” Sign Used with the Kelvin Unit

  12. Amount of Substance (mole) mole (amt of Substance) Defined • The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilograms of carbon 12. • 1 mole = 6.023x1023entities • entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.

  13. Light Brightness (candela) Luminous Intensity Defined • The luminous intensity, in a given direction, of a source that emits monochromatic radiation (one-color light) of frequency 540 x 1012 Hertz (555 nm) and that has a radiant intensity in that direction of 1/683 watt per steradian 555nmcolor • The are 4 (12.57) Steradians in a Sphere • 1 Str = 7.96% of the Sphere Surface

  14. Units Have Evolved • Candela Predecessor based on a Flame • Hence the Name • Temperature Based on Freezing points • Water • Platinum • Second Based on the Sidereal (standard) day

  15. Units Have Evolved • History of the Meter (or Metre) • One ten millionth of the distance from the North pole to the equator.  • The distance between two fine lines engraved near the ends of a platinum-iridium bar  • 1 650 763.73 wavelengths of a particular orange-red light emitted by atoms of krypton-86 (86Kr). • The length of the path traveled by light in a vacuum during a time interval of 1/299 792 458 of a second.

  16. SI Derived Units • The Seven Base Units May be Algebraically Combined to Produce “Derived Units” • e.g.: • Several DerivedUnits have SpecialUsefulness and arethus Given their OWN Names

  17. Some Derived Units

  18. Some (more) Derived Units

  19. SI prefixes – A form of ShortHand

  20. Derived Units Family Tree No Special Names

  21. Old (and Tired) Unit Sets • MKS • Stands for Meter-Kilogram-Second as the Most Common Units • Predecessor to The SI System • CGS • Means Centimeter-Gram-Second • Still Widely Used • IPS, FPM, FPH • Inch-Pound-Sec, Foot-Lb-Min, Ft-Lb-Hour

  22. Fundamental Dimension Base Unit length mass force time electric charge [Q] absolute temperature luminous intensity amount of substance foot (ft) pound (lbm) pound (lbf) second (sec) coulomb (C) degree Rankine (oR) candela (cd) mole (mol) American Engineering System, AES – Still in (declining) Use Some Are the SAME SI

  23. Conservation of Units • Principle of conservation of units: • Units on the LEFT side of an equation MUST be the SAME as those on the RIGHT side of an Equation • Then Have Dimensional Homogeneity • Needed to Prevent “Apples & Oranges” Confusion • e.g., I Buy 100 ft of Wire at One Store and 50 m at another; how much total Wire do I have? (It’s NOT “150”)

  24. Unit Conversion by Chain-Link • To Determine the Amount of Wire I have I Need to Convert to Consistent (Homogeneous) Units • Start by Thinking About the Definition of “1” • AnyThing divided by ITSELF = “1” • Now Consider a “minute” • Read as “60 Seconds per minute”

  25. Chain-Link Unit Conversion • Units can also be Multiplied and Divided in a manner similar to Numbers • This how we get, say, “Square Feet” • e.g.; Consider an 8ft x 10ft Engineer’s Cubicle in Dilbert-Land. How Much WorkSpace Does the Engineer Have? • Now Back to the Wire • Want to Know how many FEET of Wire I have in Total

  26. Chain-Link Unit Conversion cont. • Check in Table 16.8 and Find “3.2808 ft = 1meter (“3.2808 ft per meter”) • Multiply the 50m by this special Value of 1 • Can “Cancel” The Units by Division • So then the Total Wire = 264 ft

  27. Chain Link Examples • A World-Class Sprinter can Run 100m in 10s. • How Fast is this in MPH? • Gasoline In Seoul Costs 1840 Korean-Won (W) for one Liter of Regular Unleaded • How Much is this in $ per Gallon • Find Currency Exchange Rate → $1 = 1150 W

  28. Several Forms of “1” • Unit Conversion Factors • ANYTHING Divided by ItSelf = 1 

  29. Units – Exponent Properties This summary assumes that no denominators are 0 and that 00 is not considered. For any integers m and n

  30. Raising units to POWERS • Start again with 1 • Can do the SAME Thing with Units. • And 123 = 1728 so • Thus have 1728 “cubic inches” per “Cubic Foot” • What’s a “Cubic Yard” in “Cubic Feet”? • So have 27 cubic-ft per cubic-yd • NOT “9”

  31. 7 inches, Water Column • Ms. Ezersky noted that Natural Gas is delivered by PG&E to home at a pressure of 4-7 “inches of Water Column” • This is a unit of pressure, Just Like Pascals or psig • A U-Tube Manometer can Measure Pressure Differences in Inches of Water Column

  32. 7 inches, Water Column • To Calc the “in-WC” pressure we need to know some Engineering Physics • From ENGR36 • Where • ≡ Liquid SPECIFIC WEIGHT • h ≡ Liquid Column Height • For Liquid Water at Room Temperature and Pressure • Now find 7 in-WC in psig Natural Gas @ 9.5 inWC

  33. 7 inches, Water Column • Convert out the N & m

  34. White Board Examples cont. • The USA FDA recommends that Adults consume 2200 Calories per Day • What then is the “Power Rating” of a Grown Human Being? • Note that there are TWO types of “Calories” • The Amount of Heat Required to Raise the Temperature of 1 GRAM of water by 1 °C (or 1 Kelvin) • Often Called the Gram-CAL; This is what is in the Text • The Amount of Heat Required to Raise the Temperature of 1 KILOgram of water by 1 °C • Often Called the kgCAL or kiloCal; This is what you read on the side of Food Packaging

  35. Tire Pressure • Many AutoMobile Tires have a Maximum Pressure Rating of About 44 psig. • Convert 44 psi to kiloPascals (kPa)

  36. Ton of Refrigeration • During his Presentation Mr. Ian McClaren of SouthLandIndustries described the “Ice Storage” Cooling System Behind Bldg-1800. • He Noted that the Cooling Power of this system was Rated in “Tons” • What is a “Ton” of Cooling Power

  37. Ton of Refrigeration • A TON of the refrigeration is defined, roughly, as the COOLING effect of melting 2000 lbs of water ICE over a 24 HOUR Period • From PHYS4C (or ASHRAE HandBook) find that the “Latent Heat of Fusion” for ice is 333.55 kJ/kg • On WhtBoard Convert a “Ton of Refrigeration” to • kW and Btu/hr

  38. White Board Examples • A 2003 Chevy z06 corvette • Has a 5.7 Liter V8 Engine • What is the Engine Displacement in cubic-inches? • Develops 410 HP • What is the Power in Watts? • A the Maximum recommended pressure for many 65R15 tires is 44 psi (lbs per sq-inch; NOT lbs) • What is this Max Pressure in kPa?

  39. Percent (%) – What it is? • Divide the Word into Two Parts • “PER” ≡ DIVIDE • “CENT” ≡ HUNDRED • e.g., 1¢ (1 cent) = 1/100th of a Dollar ($) • Example – Home Prices • The median home price in the USA Was 13.6% Lower in 2009 than in 2008 • Meaning for every $100k of home value in 2008, the Market prices Decreased by $13.6k in 2009

  40. PerCent Notation • n% Means • But by Decimals • “n per hundred” • And • So n%

  41. PerCent as a UNIT • 100% Means • Convert 37.3% using 100%/1 = 1 • In other Words • Suggest Using Every Time

  42. Example – Covert to Decimals • Convert to Decimals a) 94% b) 7.6% • Solutions

  43. Convert TO %-Notation • Convert a) 2.39 b)0.2 c) 3/8 • Solutions Using 100%/1 = 1

  44. PerCent Change ≡ Δ% • Any Percentage Change; the “∆%”, in any quantity is expressed Relative to a BASELINE • Δ% can be either Positive or Negative • Positive Δ% → the New (or Changed) value is GREATER than the BaseLine • Negative Δ% → the New (or Changed) value is Less Than than the BaseLine • Example • My Old Car got 37mpg. My new truck gets 29mpg • Find %-Change in the Fuel Efficiency

  45. Example  % Change • The Graph at right shows the magnetic B-H behavior of an Iron-Silicon Alloy • The Graph To finch the %-Change in Flux Density when the Field Strength changes from 20 Amps per meter (A/m) to 90 A/m

  46. Example  % Change 1.60 1.22 • ThusB increases by 31.1% when H increases by 350% 90

  47. All Done for Today How toSpendtheCalories

  48. Tire Pressure: 44 psi → kPa

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