Chapter 6
Units
&
Dimensions
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
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.
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
Engineering Metrology
Measurement of dimensions Length Thickness Diameter Taper Angle Flatness profiles
Measurement Standard
Inch, foot; based on human body4000 B.C. Egypt; King’s
Elbow=0.4633 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
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 Paris39.37 in
In 1960, 1,650,763.73 wave length in vacuum of the orange light given off by electrically excited krypton 86.
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)
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?
unitnumber
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
CBAD All have identical dimensions
How do dimensions behave in mathematical formulae?
Rule 2 - Dimensions obey rules of multiplication and division
L][
][L[M]
[L]][T
][T[M]
2
2
2
C
ABD
How do dimensions behave in mathematical formulae?
Rule 3 - In scientific equations, the arguments of“transcendental functions” must be dimensionless.
xDxB
xCxA
3 )exp(
)sin( )ln(
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
··· !3!2
132
xx
xex
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
22 bBbB3
hV
.LLLLL
L 32223
1
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?
cba Lgmkp
2/1cb00L][
2/10201T][
0000M][
L][T
L[M]T][ c
2a
c
bb
aa
b
g
LkpLgkmp 2/12/10
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
amF
2T
LMF
[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
amF
cg
amF
ConversionFactor
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
gc=32.147 lbm.ft/lbf.s2
Example of Noncoherent Unit Systems
If a child weighs 50 pounds, we normally say its weight is 50.0 lbm
lbf
ftlbmslbf
sft
lbmg
mF L 0.50
*174.32*
174.32
0.50g
2
2
c
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
lbf
ftlbmslbf
sft
lbmg
mF L 6.13
*174.32*
72.8
0.50g
2
2
c
[F] [M] [L] [T]
Noncoherent × × × ×
× = defined unit— = derived unit
amF
2T
LMF
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
F=ma/gc; if we use slug for mass gc= 1.0 slug/lbf*1.0 ft/s2
1 slug=32.147 lbm1 slug times 1 ft/ s2 gives 1 lbf1 lbm times 32.147 ft/ s2 gives 1 lbf1 kg times 1 m/ s2 gives 1 N
gc= 1.0 kg/N*1.0 m/s2
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 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-133 atom”
Length or “the length of the path traveled
Distance: by light in vacuum during a time
(meter) interval of 1/299792458 seconds”
Fundamental Units (SI)
Laser1 m
photon
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”
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.
Pressure
Temperature273.16 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 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”
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
c
d
The International System of Units (SI)
Prefix Decimal Multiplier Symbol
deka
hecto
kilo
mega
Giga
Tera
Peta
exa
10+1
10+2
10+3
10+6
10+9
10+12
10+15
10+18
da
h
k
M
G
T
P
E
(SI)Force = (mass) (acceleration)
2s
m· kg 1 N 1
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 /ftslb 2f
(USCS)Force = (mass) (acceleration)
2f ft/sslug1lb1
American Engineering System of Units (AES)
Fundamenal Dimension Base Unit
length [L]
mass [m]
force [F]
time [T]
electric change [Q]
absolute temperature [
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)
2mf ft/slb1lb1
c
ma
gF
2f
m
slb
ftlb32.174
ft/s2lbm
lbf
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)
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”
Rules for Using SI Units
There should be no space between the prefix and the unit symbols Km (not k m) F (not 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
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
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)
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)
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
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
s
m
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
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
Conversions Between Systems of Units
m 0.3048 ft 1
F factor conversion1m 0.3048
ft 1
F factor conversion1ft 1
m 0.3048
m 524.1ft 1
m 0.3048 ft 5 ft 5 F
22
222 m 4676.0ft 1
m 0.3048 ft 5 ft 5
F
Temperature Scale vs Temperature Interval
32oF
212oF
T = 212oF - 32oF=180 oF
Scale Interval
Temperature Conversion
KR32CF ooo 8.18.1
R1.8
1K32F
1.8
1C ooo
Temperature Scale
Temperature Interval Conversion Factors
K
C 1
R
F 1
K
R 8.1
C
F 8.1 o
o
oo
o
o
F
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?
Team Exercise 2
Pressure loss due to pipe friction
p = pressure loss (Pa)
d = pipe diameter (m)
f = friction factor (dimensionless)
= fluid density (kg/m3)
L = pipe length (m)
v = fluid velocity (m/s)
(1) Show equation is dimensionally homogeneous
d
v L f2p
2
Team Exercise 2 (con’t)
(2) Find p (Pa) for d = 2 in, f = 0.02, = 1 g/cm3, L = 20 ft, & v = 200 ft/min
(3) Using AES units, find p (lbf/ft2) for d = 2 in, f = 0.02, = 1 g/cm3, L = 20 ft, & v = 200 ft/min
Formula ConversionsSome 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)
T49.02c
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 sameT
c02.49
2/1o2/1o Rsft
Rsft
02.49T
c
Formula ConversionsStep 3 - Convert the units
So
where
c = speed of sound (m/s)
T = temperature (K)
1/2
2/1o
2/1o2/1o s·K
m04.20
K
R 8.1
ft
m 0.3048
Rs·
ft02.49
Rsft
02.49
T20.05c
F
Team Exercise 3The flow of water over a weir can be
computedby:
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.