class 9.2 units & dimensions. objectives zknow the difference between units and dimensions...
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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)
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
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 (see Table 14.3)
[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
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 []
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”
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”
(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.
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”
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)
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
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
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 2Pressure 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
(2) Find p (Pa) for
d = 2 in, f = 0.02, = 1 g/cm3, L = 20 ft, & v = 200 ft/min
d
v L f2p
2
Pair Exercise 2 (con’t)
(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
Pair 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.