Chapter 6 Units & Dimensions. Objectives zKnow the difference between units and dimensions...

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


Rule 2 - Dimensions obey rules of multiplication and division

L][

][L[M]

[L]][T

][T[M]

2

2

2

C

ABD


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)


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”


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”


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


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”


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”


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


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)


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”


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


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


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)


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)


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


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


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


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.

Chapter 6 Units & Dimensions. Objectives zKnow the difference between units and dimensions...

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Transcript of Chapter 6 Units & Dimensions. Objectives zKnow the difference between units and dimensions...