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Chabot Mathematics Using UnitsOf Measure Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@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. • 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.
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/
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 measure. • The 1960 GCWM Specified Seven well-defined “Base” units which, by convention, are regarded as DIMENSIONALLY INDEPENDENT http://www.bipm.org/en/si/
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
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.”
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
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
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)
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
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.
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
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
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.
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
Derived Units Family Tree No Special Names
Old (and Tired) Unit Sets • MKS • Stands for Meter-Kilogram-Second in 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
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
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”)
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”
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
Chain-Link Unit Conversion cont. • Check in Table 16.8 and Find “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
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
Several Forms of “1” • Unit Conversion Factors • ANYTHING Divided by ItSelf = 1
Units – Exponent Properties This summary assumes that no denominators are 0 and that 00 is not considered. For any integers m and n
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”
7 inches, Water Column • An ENGR10 Guests Speaker 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
7 inches, Water Column • To Calc the “in-WC” pressure we need to know some Engineering Physics • From ENGR36 • Where • γ ≡ 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
7 inches, Water Column • Convert out the N & m
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
Tire Pressure • Many AutoMobile Tires have a Maximum Pressure Rating of About 44 psig. • Convert 44 psi to kiloPascals (kPa)
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
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
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?
All Done for Today How toSpendtheCalories