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

Chabot Mathematics. §1.7 SciNotat Using Units. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. MTH 55. 1.6. Review §. Any QUESTIONS About §1.6 → Exponent Properties Any QUESTIONS About HomeWork §1.6 → HW-02. Scientific Notation.

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

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  1. Chabot Mathematics §1.7 SciNotatUsing Units Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. MTH 55 1.6 Review § • Any QUESTIONS About • §1.6 → Exponent Properties • Any QUESTIONS About HomeWork • §1.6 → HW-02

  3. Scientific Notation • Scientific notation for a number is an expression of the type N× 10m • Where: N is at least 1 but less than 10 (that is, 1 ≤ N < 10), • N is expressed in decimal notation • m is an integer.

  4. Scientific Notation • Scientific notation for a number is an expression of the type N× 10m • Note that when • m is positive the decimal point moves rightm places in decimal notation • m is negative, the decimal pointmoves left |m| places.

  5. Example  Scientific Notation • Example - Convert to decimal notation: a) 3.842  106 b) 5.3  10−7 • Solution: a) Since the exponent is positive, the decimal point moves right6 places. 3.842000 → 3.842106 = 3,842,000 b) Since the exponent is negative, the decimal point moves left7 places. 0.0000005.3 → 5.310−7 = 0.00000053

  6. Example  Scientific Notation • Write in scientific notation: a) 94,000 b) 0.0423 • Solution a) We need to find m such that 94,000 = 9.4  10m. • This requires moving the decimal point 4 places to the right. • 94,000 = 9.4 104

  7. Example  Scientific Notation • Write in scientific notation: a) 94,000b) 0.0423 • Solution b) To change 0.0423 to 4.23 we move the decimal point 2 places to the left. • 0.0423 = 4.23 10–2

  8. Multiplying and Dividing Using Scientific Notation • Products and quotients of numbers written in scientific notation are found using the rules for exponents. Example - Simplify: (1.7108)(2.210−5) Solution(1.7  108)(2.2  10−5) = (1.7  2.2)  (108  10−5) = 3.74  108 +(−5) = 3.74  103

  9. Example  Divide • Simplify (6.2  10−9)  (8.0  108) • Solution (6.210−9)  (8.0108) =

  10. Multiply & Divide Summary • If Multiplying in Scientific Notation, then • MULTIPLY Decimal Numbers • ADD Exponent Numbers • If Dividing in Scientific Notation, then • DIVIDE Decimal Numbers • SUBTRACT Exponent Numbers

  11. Scientific Notation Procedure • Move the decimal point to the right or left until you have a number that is greater than or equal to 1, but less than 10. • Count how many places you moved the decimal point. This number will become the absolute value of the exponent. • If you moved the decimal point to the left, the exponent will be positive. • If you moved the decimal point to the right, make the exponent negative.

  12. Left↔Right? Top↨Bottom? What??? • When deciding on the SIGN for the Exponent in Scientific Notation  • If the Number is ≥10, then theExponent is POSTIVE • If the Number is <1, then the Exponent is NEGATIVE • If the Number is ≥1 & <10, then the Exponent is ZERO • i.e., NO “x10n” needed

  13. Write in Scientific Notation: 1043 2.5 0.000495 More Examples • Scientific Notation Solutions • 1.043103 • 2.5100 = 2.51 = 2.5 • 4.9510−4 • The decimal is to the right of the 3. Move it LEFT 3 places. • This number is already greater than or equal to one and less than 10. Therefore, the decimal does NOT have to be moved and the exponent will be 0 • Move the decimal RIGHT 4 places.

  14. As you will learn when you take CHEM1A & ENGR45 all matter is made of VERY small particles called ATOMS Atoms are, in turn, composed of SUB-atomic Particles The Primary SubAtomic Particles and their masses Protons → 1.6710−27 kg Neutrons → 1.6710−27 kg Electrons → 9.1110−31 kg Example  Mass of an Atom

  15. Example  Mass of 107Ag • Now take the Metal Silver (Chem Symbol Ag). • The 107Ag atom “Isotope” contains • 47 Protons • 47 Electrons • 60 Neutrons • Find the Mass of a 107Ag atom

  16. Example  Mass of 107Ag • Find Total PROTON Mass • Find Total ELECTRON Mass • Find Total NEUTRON Mass

  17. Example  Mass of 107Ag • Add the Total Masses of the all the SubAtomic particles

  18. Chabot Mathematics Chp1 ExtraUsing Units Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  19. Physical Quantities • Anything that we can “Feel” or “See” or “Sense” can be MEASURED. These Things are PHYSICAL Quantities • e.g.; Time, Temperature, Length, Angle • To “Measure” a physical quantity We need a “Ruler” that describes the “Size” of the Quantity. This “Sizing” leads to the concept of UNITS

  20. 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 People 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.

  21. SI System of Units • Système International d'Unités (International System of Units) • A Completely Consistent Set of Basic Units • Requires NO Conversion factors • e.g., 18 inches = 1.5 feet • Defined by UNCHANGING Physical Phenomena • Except for one... http://www.bipm.org/en/si/

  22. 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

  23. 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

  24. 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.”

  25. 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

  26. 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 vacuum, would produce between these conductors a force equal to 2 x 10−7 Newton per metre of length. • What’s a Newton?→ 1kg-m/(s2)

  27. 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

  28. 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 kilogram of carbon 12. • 1 mol = 6.023x1023entities • entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.

  29. 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 (λ = 555.5 nm) and that has a radiant intensity in that direction of 1/683 watt per steradian • The are 4 (12.57) Steradians in a sphere • 1 Str = 7.96% of the Sphere Surface

  30. 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

  31. 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.

  32. SI Derived Units • The Seven Base Units May be Arithmetically Combined to Produce “Derived Units” • e.g.: • Several DerivedUnits have SpecialUsefulness AndGiven their OWNNames

  33. Some Derived Units

  34. Some (more) Derived Units

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

  36. 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

  37. 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”

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

  39. Chain-Link Unit Conversion • Also Units can 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

  40. Chain-Link Unit Conversion cont. • Check on the Internet and Find that there are 3.2808ft in one meter • Multiply the 50m by this special Value of 1 • Can “Cancel” The Units by Division • So then the Total Wire = 100ft + 164ft = 264 ft

  41. Chain Link Examples • A World-Class Sprinter can Run 100m in 10s. • How Fast is this in MPH? • Gasoline In Hamburg Germany Costs 1.10 € for one Liter of Regular Unleaded • How Much is this in $ per Gallon • Find Currency Exchange Rate → $1 = 0.787 €

  42. Some Unit Conversions • See Also http://www.onlineconversion.com/

  43. WhiteBoard Work • Problems From §1.7 Exercise Set • 64, 66, 68 • “Seward’s Folly” ≡ 1868 Purchase of Alaska from RussiaFor $7.2M

  44. White Board Example • The USA FDA recommends that Adults consume 2200 Calories per Day • What then is the “Power Rating” of a Grown Human Being? • Note that 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; used in the CGS system • The Amount of Heat Required to Raise the Temperature of 1 KILOgram of water by 1 °C • Often Called the KgCAL; This is what you read on the side of Food Packaging

  45. White Board Examples • A 1966 Dodge Hemi Engine • 426 Cubic Inch V8 • What is the Engine Displacement in Litres? • Develops 425 HP • What is the Power in Watts? • The 2006 Toyota Prius Hybrid Synergy Drive System has a Torque rating of 400 Newton-Meters (Nm) • What is this Torque in Ft-Lbs?

  46. All Done for Today More Infoon SilverIsotopes • http://www.webelements.com/webelements/elements/text/Ag/isot.html

  47. All Done for Today • 16 tablespoons = 1 cup • 12 tablespoons = 3/4 cup • 10 tablespoons + 2 teaspoons = 2/3 cup • 8 tablespoons = 1/2 cup • 6 tablespoons = 3/8 cup • 5 tablespoons + 1 teaspoon = 1/3 cup • 4 tablespoons = 1/4 cup • 2 tablespoons = 1/8 cup • 1 tablespoon = 1/16 cup • 2 cups = 1 pint • 2 pints = 1 quart • 3 teaspoons = 1 tablespoon • 48 teaspoons = 1 cup CookingConversions

  48. Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

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