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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 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

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.

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

How do dimensions behave in mathematical formulae?

Rule 2 - Dimensions obey rules of multiplication

and division

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

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

Dimensional Analysis

Pendulum - What is the period?

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

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

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

Example of Noncoherent Unit Systems

- If a child weighs 50 pounds, we normally say its

weight is 50.0 lbm

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

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)

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

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)

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

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

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.

Water Phase Diagram

Pressure

Temperature

273.16 K

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 m c d

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

(SI)Force (mass) (acceleration)

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

(USCS)Force (mass) (acceleration)

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)

(AES)Force (mass) (acceleration)

lbm

ft/s2

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)
- 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/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

Temperature Scale vs Temperature Interval

DT 212oF - 32oF180 oF

Scale

Interval

Temperature Conversion

Temperature Scale

Temperature Interval Conversion Factors

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
- 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 (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

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)

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

Formula Conversions

- Step 3 - Convert the units
- So
- 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.