Chapter 6 Units & Dimensions

Objectives zKnow the difference between units and dimensions zUnderstand the SI, USCS (U.S. Customary System, or British Gravitational System), and AES (American Engineering) systems of units zKnow the SI prefixes from nano- to giga- zUnderstand and apply the concept of dimensional homogeneity

Objectives zWhat is the difference between an absolute and a gravitational system of units? zWhat is a coherent system of units? zApply dimensional homogeneity to constants and equations.

Introduction zFrance in 1840 legislated official adoption of the metric system and made its use be mandatory zIn U.S., in 1866, the metric system was made legal, but its use was not compulsory

Engineering Metrology zMeasurement of dimensions yLength yThickness yDiameter yTaper yAngle yFlatness yprofiles

Measurement Standard zInch, foot; based on human body z4000 B.C. Egypt; King’s Elbow= m, 1.5 ft, 2 handspans, 6 hand-widths, 24 finger-thickness zAD 1101 King Henry I  yard ( m) from his nose to the tip of his thumb z1528 French physician J. Fernel  distance between Paris and Amiens

Measurement Standard z1872, Meter (in Greek, metron to measure)- 1/10 of a millionth of the distance between the North Pole and the equator zPlatinum (90%)-iridium (10%) X- shaped bar kept in controlled condition in Paris  in zIn 1960, 1,650, wave length in vacuum of the orange light given off by electrically excited krypton 86.

Dimensions & Units zDimension - abstract quantity (e.g. length) yDimensions are used to describe physical quantities yDimensions are independent of units zUnit - a specific definition of a dimension based upon a physical reference (e.g. meter)

What does a “unit” mean? Rod of unknown length Reference: Three rods of 1-m length The unknown rod is 3 m long. How long is the rod? unit number The number is meaningless without the unit!

How do dimensions behave in mathematical formulae? Rule 1 - All terms that are added or subtracted must have same dimensions All have identical dimensions

How do dimensions behave in mathematical formulae? Rule 2 - Dimensions obey rules of multiplication and division

How do dimensions behave in mathematical formulae? Rule 3 - In scientific equations, the arguments of “transcendental functions” must be dimensionless. x must be dimensionless Exception - In engineering correlations, the argument may have dimensions Transcendental Function - Cannot be given by algebraic expressions consisting only of the argument and constants. Requires an infinite series

Dimensionally Homogeneous Equations An equation is said to be dimensionally homogeneous if the dimensions on both sides of the equal sign are the same.

Dimensionally Homogeneous Equations B b h Volume of the frustrum of a right pyramid with a square base

Dimensional Analysis g m L p Pendulum - What is the period?

Absolute and Gravitational Unit Systems zAbsolute system yDimensions used are not affected by gravity yFundamental dimensions L,T,M zGravitational System yWidely used used in engineering yFundamental dimensions L,T,F

[F] [M] [L] [T] Absolute — × × × Gravitational × — × × × = defined unit — = derived unit Absolute and Gravitational Unit Systems

Coherent Systems - equations can be written without needing additional conversion factors Coherent and Noncoherent Unit Systems Noncoherent Systems - equations need additional conversion factors Conversion Factor

Noncoherent Unit Systems zOne pound-force (lbf) is the effort required to hold a one pound-mass elevated in a gravitational field where the local acceleration of gravity is ft/s 2 zConstant of proportionality g c should be used if slug is not used for mass zg c = lbm.ft/lbf.s 2

Example of Noncoherent Unit Systems zIf a child weighs 50 pounds, we normally say its weight is 50.0 lbm

Example of Noncoherent Unit Systems zIf a child weighs 50 pounds, on a planet where the local acceleration of gravity is 8.72 ft/s 2

[F] [M] [L] [T] Noncoherent × × × × × = defined unit — = derived unit Noncoherent Systems The noncoherent system results when all four quantities are defined in a way that is not internally consistent (both mass and weight are defined historically)

Coherent System zF=ma/g c ; if we use slug for mass yg c = 1.0 slug/lbf*1.0 ft/s 2 z1 slug= lbm z1 slug times 1 ft/ s 2 gives 1 lbf z1 lbm times ft/ s 2 gives 1 lbf z1 kg times 1 m/ s 2 gives 1 N yg c = 1.0 kg/N*1.0 m/s 2

The International System of Units (SI) Fundamental Dimension Base Unit length [L] mass [M] time [T] electric current [A] absolute temperature [  ] luminous intensity [l] amount of substance [n] meter (m) kilogram (kg) second (s) ampere (A) kelvin (K) candela (cd) mole (mol)

The International System of Units (SI) Supplementary DimensionBase Unit plane angle solid angle radian (rad) steradian (sr)

Fundamental Units (SI) Mass: “a cylinder of platinum-iridium (kilogram) alloy maintained under vacuum conditions by the International Bureau of Weights and Measures in Paris”

Fundamental Units (SI) Time: “the duration of 9,192,631,770 periods (second) of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom”

Length or “the length of the path traveled Distance: by light in vacuum during a time (meter) interval of 1/ seconds” Fundamental Units (SI) Laser 1 m photon t = 0 s t = 1/ s

Fundamental Units (SI) Electric “that constant current which, if Current:maintained in two straight parallel (ampere)conductors of infinite length, of negligible circular cross section, and placed one meter apart in a vacuum, would produce between these conductors a force equal to 2 × newtons per meter of length”

Fundamental Units (SI) Temperature: The kelvin unit is 1/ of the (kelvin)temperature interval from absolute zero to the triple point of water. Pressure Temperature K Water Phase Diagram

Fundamental Units (SI) AMOUNT OF “the amount of a substance that SUBSTANCE: contains as many elementary enti- (mole) ties as there are atoms in kilograms of carbon 12”

Fundamental Units (SI) LIGHT OR“the candela is the luminous LUMINOUSintensity of a source that emits INTENSITY:monochromatic radiation of (candela)frequency 540 × Hz and that has a radiant intensity of 1/683 watt per steradian.“ See Figure 13.5 in Foundations of Engineering

Supplementary Units (SI) PLANE“the plane angle between two radii ANGLE:of a circle which cut off on the (radian)circumference an arc equal in length to the radius:

Supplementary Units (SI) SOLID“the solid angle which, having its ANGLE:vertex in the center of a sphere, (steradian)cuts off an area of the surface of the sphere equal to that of a square with sides of length equal to the radius of the sphere”

The International System of Units (SI) PrefixDecimal MultiplierSymbol Atto Femto pico nano micro milli centi deci afpnmcdafpnmcd

The International System of Units (SI) PrefixDecimal MultiplierSymbol deka hecto kilo mega Giga Tera Peta exa da h k M G T P E

(SI) Force = (mass) (acceleration)

U.S. Customary System of Units (USCS) Fundamenal DimensionBase Unit length [L] force [F] time [T] foot (ft) pound (lb) second (s) Derived DimensionUnitDefinition mass [FT 2 /L]slug

(USCS) Force = (mass) (acceleration)

American Engineering System of Units (AES) Fundamenal DimensionBase Unit length [L] mass [m] force [F] time [T] electric change [Q] absolute temperature [  luminous intensity [l] amount of substance [n] foot (ft) pound (lb m ) pound (lb f ) second (sec) coulomb (C) degree Rankine ( o R) candela (cd) mole (mol)

(AES) Force = (mass) (acceleration) ft/s 2 lb m lb f

Rules for Using SI Units zPeriods are never used after symbols yUnless at the end of the sentence ySI symbols are not abbreviations zIn lowercase letter unless the symbol derives from a proper name ym, kg, s, mol, cd (candela) yA, K, Hz, Pa (Pascal), C (Celsius)

Rules for Using SI Units zSymbols rather than self-styles abbreviations always should be used yA (not amp), s (not sec) zAn s is never added to the symbol to denote plural zA space is always left between the numerical value and the unit symbol y43.7 km (not 43.7km) y0.25 Pa (not 0.25Pa) yException; 5 0 C, 5’ 6”

Rules for Using SI Units zThere should be no space between the prefix and the unit symbols yKm (not k m)   F (not  F) zWhen writing unit names, lowercase all letters except at the beginning of a sentence, even if the unit is derived from a proper name yFarad, hertz, ampere

Rules for Using SI Units zPlurals are used as required when writing unit names yHenries (H; henry) yExceptions; lux, hertz, siemens zNo hyphen or space should be left between a prefix and the unit name yMegapascal (not mega-pascal) yExceptions; megohm, kilohm, hetare

Rules for Using SI Units zThe symbol should be used in preference to the unit name because unit symbols are standardized yExceptions; ten meters (not ten m) y10 m (not 10 meters)

Rules for Using SI Units zWhen writing unit names as a product, always use a space (preferred) or a hyphen ynewton meter or newton-meter zWhen expressing a quotient using unit names, always use the word per and not a solidus (slash mark /), which is reserved for use with symbols ymeter per second (not meter/second)

Rules for Using SI Units zWhen writing a unit name that requires a power, use a modifier, such as squared or cubed, after the unit name ymillimeter squared (not square millimeter) zWhen expressing products using unit symbols, the center dot is preferred yN.m for newton meter

Rules for Using SI Units zWhen denoting a quotient by unit symbols, any of the follow methods are accepted form ym/s ym.s -1 yor yM/s 2 is good but m/s/s is not yKg.m 2 /(s 3.A) or kg.m 2.s -3.A -1 is good, not kg.m 2 /s 3 /A

Rules for Using SI Units zTo denote a decimal point, use a period on the line. When expressing numbers less than 1, a zero should be written before the decimal y15.6 y0.93

Rules for Using SI Units zSeparate the digits into groups of three, counting from the decimal to the left or right, and using a small space to separate the groups y y y7 434 y

Conversions Between Systems of Units

Temperature Scale vs Temperature Interval 32 o F 212 o F  T = 212 o F - 32 o F=180 o F Scale Interval

Temperature Conversion Temperature Scale Temperature Interval Conversion Factors

Team Exercise 1 zThe force of wind acting on a body can be computed by the formula: F = C d V 2 A where: F = wind force (lb f ) C d = drag coefficient (no units) V = wind velocity (mi/h) A = projected area(ft 2 ) zTo keep the equation dimensionally homogeneous, what are the units of ?

Team Exercise 2 Pressure loss due to pipe friction  p = pressure loss (Pa) d = pipe diameter (m) f = friction factor (dimensionless)  = fluid density (kg/m 3 ) L = pipe length (m) v = fluid velocity (m/s) (1) Show equation is dimensionally homogeneous

Team Exercise 2 (con’t) (2) Find  p (Pa) for d = 2 in, f = 0.02,  = 1 g/cm 3, L = 20 ft, & v = 200 ft/min (3) Using AES units, find  p (lb f /ft 2 ) for d = 2 in, f = 0.02,  = 1 g/cm 3, L = 20 ft, & v = 200 ft/min

Formula Conversions Some formulas have numeric constants that are not dimensionless, i.e. units are hidden in the constant. As an example, the velocity of sound is expressed by the relation, where c = speed of sound (ft/s) T = temperature ( o R)

Formula Conversions Convert this relationship so that c is in meters per second and T is in kelvin. Step 1 - Solve for the constant Step 2 - Units on left and right must be the same

Formula Conversions Step 3 - Convert the units So where c = speed of sound (m/s) T = temperature (K) F

Team Exercise 3 The flow of water over a weir can be computed by: Q = 5.35LH 3/2 where: Q = volume of water (ft 3 /s) L = length of weir(ft) H = height of water over weir (ft) Convert the formula so that Q is in gallons/min and L and H are measured in inches.