Class 9.2 Units & Dimensions

Objectives Know the difference between units and dimensions Understand the SI, USCS, and AES systems of units Know the SI prefixes from nano- to giga- Understand and apply the concept of dimensional homogeneity

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

RAT 13

Dimensions & Units Dimension - abstract quantity (e.g. length) Unit - a specific definition of a dimension based upon a physical reference (e.g. meter)

The unknown rod is 3 m long. What does a “unit” mean? How long is the rod? Rod of unknown length Reference: Three rods of 1-m length The unknown rod is 3 m long. 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 Volume of the frustrum of a right pyramid with a square base B b h

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

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

Coherent and Noncoherent Unit Systems Coherent Systems - equations can be written without needing additional conversion factors Noncoherent Systems - equations need additional conversion factors Conversion Factor (see Table 14.3)

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

Examples of Unit Systems See Table 14.1

The International System of Units (SI) Fundamental Dimension Base Unit length [L] mass [M] time [T] electric current [A] absolute temperature [q] 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 Dimension Base 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-113 atom”

Fundamental Units (SI) Length or “the length of the path traveled Distance: by light in vacuum during a time (meter) interval of 1/299792458 seconds” photon Laser 1 m t = 0 s t = 1/299792458 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 × 10-7 newtons per meter of length” (see Figure 13.3 in Foundations of Engineering)

Fundamental Units (SI) Temperature: The kelvin unit is 1/273.16 of the (kelvin) temperature interval from absolute zero to the triple point of water. Water Phase Diagram Pressure Temperature 273.16 K

Fundamental Units (SI) AMOUNT OF “the amount of a substance that SUBSTANCE: contains as many elementary enti- (mole) ties as there are atoms in 0.012 kilograms of carbon 12”

Fundamental Units (SI) LIGHT OR “the candela is the luminous LUMINOUS intensity of a source that emits INTENSITY: monochromatic radiation of (candela) frequency 540 × 1012 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”

Derived Units See Foundations, Table 13.4 Most important... J, N, Hz, Pa, W, C, V

The International System of Units (SI) Prefix Decimal Multiplier Symbol nano micro milli centi deci deka hecto kilo mega giga 10-9 10-6 10-3 10-2 10-1 10+1 10+2 10+3 10+6 10+9 n m c d da h k M G

(SI) Force = (mass) (acceleration)

U.S. Customary System of Units (USCS) Fundamenal Dimension Base Unit length [L] force [F] time [T] foot (ft) pound (lb) second (s) Derived Dimension Unit Definition mass [FT2/L] slug

(USCS) Force = (mass) (acceleration)

American Engineering System of Units (AES) Fundamenal Dimension Base Unit length [L] mass [m] force [F] time [T] electric change [Q] absolute temperature [q] luminous intensity [l] amount of substance [n] foot (ft) pound (lbm) pound (lbf) second (sec) coulomb (C) degree Rankine (oR) candela (cd) mole (mol)

(AES) Force = (mass) (acceleration) lbm ft/s2 lbf

Conversions Between Systems of Units

Temperature Scale vs Temperature Interval 212oF 32oF DT = 212oF - 32oF=180 oF Scale Interval

Temperature Conversion Temperature Scale Temperature Interval Conversion Factors

Pairs Exercise 1 The force of wind acting on a body can be computed by the formula: F = 0.00256 Cd V2 A where: F = wind force (lbf) Cd= drag coefficient (no units) V = wind velocity (mi/h) A = projected area(ft2) To keep the equation dimensionally homogeneous, what are the units of 0.00256?

Pair Exercise 2 Pressure loss due to pipe friction Dp = pressure loss (Pa) d = pipe diameter (m) f = friction factor (dimensionless) r = fluid density (kg/m3) L = pipe length (m) v = fluid velocity (m/s) (1) Show equation is dimensionally homogeneous (2) Find D p (Pa) for d = 2 in, f = 0.02, r = 1 g/cm3, L = 20 ft, & v = 200 ft/min

Pair Exercise 2 (con’t) (3) Using AES units, find D p (lbf/ft2) for d = 2 in, f = 0.02, r = 1 g/cm3, 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 (oR)

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

Pair Exercise 3 The flow of water over a weir can be computed by: Q = 5.35LH3/2 where: Q = volume of water (ft3/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.