Transcript Fundamental Units

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 body
4000 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%) Xshaped 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.
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?
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
D  A B C
All have identical dimensions
How do dimensions behave
in mathematical formulae?
Rule 2 - Dimensions obey rules of
multiplication and division
 [M]  [T 2 ] 

 2 
AB  [T ]  [L] 
D

 [L]
C
 [M] 
 2 
 [L ] 
How do dimensions behave
in mathematical formulae?
Rule 3 - In scientific equations, the arguments of
“transcendental functions” must be dimensionless.
A  ln( x)
C  sin( x)
B  exp( x)
D3
x
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
2
3
series
x
x
e  1  x   ···
2! 3!
x
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
h


B
h 2
V  B  Bb  b2
3
L 2
3
L     L  L2  L2
 1
 
       L .
3
Dimensional Analysis
Pendulum - What is the period?
pk m
[T]  [M]a
[M]
[T]
[L]
0
1
0



a
0
0
p  km g
0
a
g
b
L
 T 2 

0

2b

b
1 / 2 1 / 2
L
L
c
L
m
b
p
[L]c



0
0
c
L
 pk
g
g

a0
 b  1 / 2
 c  1 / 2
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
Absolute and Gravitational Unit Systems
F m a

L
F  M 2
T 
[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
F m a
Noncoherent Systems - equations need additional
conversion factors
a
F m
gc
Conversion
Factor
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
32.174 ft
2
gL
s
F m
 50.0lbm
 50.0lbf
2
lbf * s
gc
32.174lbm * ft
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
8.72 ft
2
gL
s
F m
 50.0lbm
 13.6lbf
2
lbf * s
gc
32.174lbm * ft
Noncoherent Systems
F m a

L
F  M 2
T 
Noncoherent
[F]
[M]
[L]
[T]
×
×
×
×
× = 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)
Coherent System
F=ma/gc; if we use slug for mass
gc= 1.0 slug/lbf*1.0 ft/s2
1
1
1
1
slug=32.147 lbm
slug times 1 ft/ s2 gives 1 lbf
lbm times 32.147 ft/ s2 gives 1 lbf
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]
meter (m)
mass [M]
kilogram (kg)
time [T]
second (s)
electric current [A]
ampere (A)
absolute temperature [q]
luminous intensity [l]
amount of substance [n]
kelvin (K)
candela (cd)
mole (mol)
The International System of
Units (SI)
Supplementary Dimension
Base Unit
plane angle
radian (rad)
solid angle
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”
Fundamental Units (SI)
Length or
Distance:
(meter)
“the length of the path traveled
by light in vacuum during a time
interval of 1/299792458 seconds”
photon
Laser
1m
t=0s
t = 1/299792458 s
Fundamental Units (SI)
Electric
Current:
(ampere)
“that constant current which, if
maintained in two straight parallel
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:
(kelvin)
The kelvin unit is 1/273.16 of the
temperature interval from absolute
zero to the triple point of water.
Water Phase Diagram
Pressure
273.16 K
Temperature
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
LUMINOUS
INTENSITY:
(candela)
“the candela is the luminous
intensity of a source that emits
monochromatic radiation of
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
ANGLE:
(radian)
“the plane angle between two radii
of a circle which cut off on the
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
10-18
a
Femto
10-15
f
pico
10-12
p
nano
10-9
n
micro
10-6
m
milli
10-3
m
centi
10-2
c
deci
10-1
d
The International System of Units (SI)
Prefix
Decimal Multiplier
Symbol
deka
10+1
da
hecto
10+2
h
kilo
10+3
k
mega
10+6
M
Giga
10+9
G
Tera
10+12
T
Peta
10+15
P
exa
10+18
E
(SI)
Force = (mass) (acceleration)
m
1 N  1 kg· 2
s
U.S. Customary System of
Units (USCS)
Fundamenal Dimension
length [L]
foot (ft)
force [F]
pound (lb)
time [T]
second (s)
Derived Dimension
mass [FT2/L]
Base Unit
Unit
slug
Definition
lb f  s2 /ft
(USCS)
Force = (mass) (acceleration)
1 lb f  1 slug  ft/s
2
American Engineering System
of Units (AES)
Fundamenal Dimension
Base Unit
length [L]
foot (ft)
mass [m]
pound (lbm)
force [F]
pound (lbf)
time [T]
second (sec)
electric change [Q]
coulomb (C)
absolute temperature [q
degree Rankine (oR)
luminous intensity [l]
candela (cd)
amount of substance [n]
mole (mol)
(AES)
Force = (mass) (acceleration)
1 lb f  1 lbm  ft/s 2
lbm
lbf
ma
F 
gc
lb m ft
32.174
lb f s2
ft/s2
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)
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
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
s
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
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
1 ft  0.3048 m
1 ft
 1  conversion factor  F
0.3048 m
0.3048 m
 1  conversion factor  F
1 ft
0.3048 m
5 ft  F  5 ft 
 1.524 m
1 ft
2
 0.3048 m 
2
5 ft  F  5 ft  

0
.
4676
m

 1 ft

2
2
2
Temperature Scale vs
Temperature Interval
212oF
32oF
DT = 212oF - 32oF=180 oF
Scale
Interval
Temperature Conversion
Temperature Scale
o
o
 
R  1.8K 
F  1.8 o C  32
1
C 
1.8
F
o
 32
o

1
K 
1.8
Temperature Interval Conversion Factors
o
o
o
o
1.8 F 1.8 R 1 F 1 C
F o

 o 
C
K
R
K
 R
o
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
2 f L r v2
Dp 
d
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
Team Exercise 2 (con’t)
(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
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,
c  49.02 T
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
c
49.02 
T
Step 2 - Units on left and right must be the same
ft
c
s
49.02 o 1/ 2 
R
T
ft
s
o 1/ 2
R
Formula Conversions
Step 3 - Convert the units
ft
1/ 2
o
ft
0.3048 m  1.8 R 
m
s
  20.04
49.02 o 1/ 2  49.02 o 1/ 2 
 
1/2
R
s· R
ft
K
s·K


So
c  20.05 T
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 = 5.35LH3/2
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.