class 9.2 units & dimensions. objectives zknow the difference between units and dimensions...

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.