Units

1
Chapter 6

• Units
• Dimensions

2
Objectives

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

3
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.

4
Introduction

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

5
Engineering Metrology

• Measurement of dimensions
• Length
• Thickness
• Diameter
• Taper
• Angle
• Flatness
• profiles

6
Measurement Standard

• Inch, foot based on human body
• 4000 B.C. Egypt Kings Elbow0.4633 m, 1.5 ft, 2
  handspans, 6 hand-widths, 24 finger-thickness
• AD 1101 King Henry I ?yard (0.9144 m) from his
  nose to the tip of his thumb
• 1528 French physician J. Fernel ?distance between
  Paris and Amiens

7
Measurement Standard

• 1872, Meter (in Greek, metron to measure)- 1/10
  of a millionth of the distance between the North
  Pole and the equator
• Platinum (90)-iridium (10) X-shaped bar kept in
  controlled condition in Paris?39.37 in
• In 1960, 1,650,763.73 wave length in vacuum of
  the orange light given off by electrically
  excited krypton 86.

8
Dimensions Units

• Dimension - abstract quantity (e.g. length)
• Dimensions are used to describe physical
  quantities
• Dimensions are independent of units
• Unit - a specific definition of a dimension based
  upon a
  
  physical
  reference (e.g. meter)

9
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!
10
How do dimensions behave in mathematical formulae?
Rule 1 - All terms that are added or subtracted
must have same dimensions
11
How do dimensions behave in mathematical formulae?
Rule 2 - Dimensions obey rules of multiplication
and division
12
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
13
Dimensionally Homogeneous Equations

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

14
Dimensionally Homogeneous Equations
Volume of the frustrum of a right pyramid with a
square base
15
Dimensional Analysis
Pendulum - What is the period?
16
Absolute and Gravitational Unit Systems

• Absolute system
• Dimensions used are not affected by gravity
• Fundamental dimensions L,T,M
• Gravitational System
• Widely used used in engineering
• Fundamental dimensions L,T,F

17
Absolute and Gravitational Unit Systems
18
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
19
Noncoherent Unit Systems

• One 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
  32.147 ft/s2
• Constant of proportionality gc should be used if
  slug is not used for mass
• gc32.147 lbm.ft/lbf.s2

20
Example of Noncoherent Unit Systems

• If a child weighs 50 pounds, we normally say its
  weight is 50.0 lbm

21
Example of Noncoherent Unit Systems

• If a child weighs 50 pounds, on a planet where
  the local acceleration of gravity is 8.72 ft/s2

22
Noncoherent Systems
defined unit derived unit
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)
23
Coherent System

• Fma/gc if we use slug for mass
• gc 1.0 slug/lbf1.0 ft/s2
• 1 slug32.147 lbm
• 1 slug times 1 ft/ s2 gives 1 lbf
• 1 lbm times 32.147 ft/ s2 gives 1 lbf
• 1 kg times 1 m/ s2 gives 1 N
• gc 1.0 kg/N1.0 m/s2

24
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)
25
The International System of Units (SI)
Supplementary Dimension
Base Unit
plane angle solid angle
radian (rad) steradian (sr)
26
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

27
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

28
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
29
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

30
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
31
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

32
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
33
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

34
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

35
The International System of Units (SI)
Prefix
Decimal Multiplier
Symbol
Atto Femto pico nano micro milli centi deci
10-18 10-15 10-12 10-9 10-6 10-3 10-2 10-1
a f p n m m c d
36
The International System of Units (SI)
Prefix
Decimal Multiplier
Symbol
deka hecto kilo mega Giga Tera Peta exa
101 102 103 106 109 1012 1015 1018
da h k M G T P E
37
(SI)Force (mass) (acceleration)
38
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
39
(USCS)Force (mass) (acceleration)

40
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)
41
(AES)Force (mass) (acceleration)

lbm
ft/s2
lbf
42
Rules for Using SI Units

• Periods are never used after symbols
• Unless at the end of the sentence
• SI symbols are not abbreviations
• In lowercase letter unless the symbol derives
  from a proper name
• m, kg, s, mol, cd (candela)
• A, K, Hz, Pa (Pascal), C (Celsius)

43
Rules for Using SI Units

• Symbols rather than self-styles abbreviations
  always should be used
• A (not amp), s (not sec)
• An s is never added to the symbol to denote
  plural
• A space is always left between the numerical
  value and the unit symbol
• 43.7 km (not 43.7km)
• 0.25 Pa (not 0.25Pa)
• Exception 50C, 5 6

44
Rules for Using SI Units

• There should be no space between the prefix and
  the unit symbols
• Km (not k m)
• mF (not m F)
• When writing unit names, lowercase all letters
  except at the beginning of a sentence, even if
  the unit is derived from a proper name
• Farad, hertz, ampere

45
Rules for Using SI Units

• Plurals are used as required when writing unit
  names
• Henries (H henry)
• Exceptions lux, hertz, siemens
• No hyphen or space should be left between a
  prefix and the unit name
• Megapascal (not mega-pascal)
• Exceptions megohm, kilohm, hetare

46
Rules for Using SI Units

• The symbol should be used in preference to the
  unit name because unit symbols are standardized
• Exceptions ten meters (not ten m)
• 10 m (not 10 meters)

47
Rules for Using SI Units

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

48
Rules for Using SI Units

• When writing a unit name that requires a power,
  use a modifier, such as squared or cubed, after
  the unit name
• millimeter squared (not square millimeter)
• When expressing products using unit symbols, the
  center dot is preferred
• N.m for newton meter

49
Rules for Using SI Units

• When denoting a quotient by unit symbols, any of
  the follow methods are accepted form
• m/s
• m.s-1
• or
• M/s2 is good but m/s/s is not
• Kg.m2/(s3.A) or kg.m2.s-3.A-1 is good, not
  kg.m2/s3/A

50
Rules for Using SI Units

• To denote a decimal point, use a period on the
  line. When expressing numbers less than 1, a
  zero should be written before the decimal
• 15.6
• 0.93

51
Rules for Using SI Units

• Separate the digits into groups of three,
  counting from the decimal to the left or right,
  and using a small space to separate the groups
• 6.513 824
• 76 851
• 7 434
• 0.187 62

52
Conversions Between Systems of Units
53
Temperature Scale vs Temperature Interval
DT 212oF - 32oF180 oF
Scale
Interval
54
Temperature Conversion
Temperature Scale
Temperature Interval Conversion Factors
55
Team 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?

56
Team 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

57
Team Exercise 2 (cont)

• (2) Find Dp (Pa) for d 2 in, f 0.02, r 1
  g/cm3, L 20 ft, v 200 ft/min
• (3) Using AES units, find Dp (lbf/ft2) for d 2
  in, f 0.02, r 1 g/cm3, L 20 ft, v 200
  ft/min

58
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)

59
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

60
Formula Conversions

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

F
61
Team 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.