1. Class 9.2
Units
&
Dimensions

2. 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

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. RAT 13

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

6. 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!

7. How do dimensions behave in mathematical formulae?
Rule 1 - All terms that are added or subtracted must have same dimensions
All have identical dimensions

8. How do dimensions behave in mathematical formulae?
Rule 2 - Dimensions obey rules of multiplication and division

9. 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

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

11. Dimensionally Homogeneous Equations
Volume of the frustrum of a right pyramid
with a square base B b h

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

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

14. 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)

15. 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

16. Examples of Unit Systems
See Table 14.1

17. 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)

18. The International System of Units (SI)
Supplementary Dimension
Base Unit
plane angle
solid angle
radian (rad)
steradian (sr)

19. 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”

20. 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”

21. Fundamental Units (SI)
Length or “the length of the path traveled
Distance: by light in vacuum during a time
(meter) interval of 1/ seconds”
photon
Laser
1 m
t = 0 s
t = 1/ s

22. 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)

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

24. 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”

25. 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

26. 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:

27. 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”

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

29. 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

30. (SI) Force = (mass) (acceleration)

31. 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

32. (USCS) Force = (mass) (acceleration)

33. 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)

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

35. Conversions Between Systems of Units

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

37. Temperature Conversion
Temperature Scale
Temperature Interval Conversion Factors

38. Pairs Exercise 1
The force of wind acting on a body can be computed by the formula:
F = 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 ?

39. 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

40. 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

41. 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)

42. 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

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

44. 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.