[email protected] MTH1516_Sp14_Using_Units.pptx 1 Bruce Mayer, PE Chabot College Mathematics...

[email protected] • MTH1516_Sp14_Using_Units.pptx1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Units &Conversio

ns



Units Introduction People measure quantities

through comparisons with standards.

Every measured quantity has an associated “unit” Which is the name of the Standard.

Need to define sensible and practical "units" and "standards" that scientists & engineers everywhere can agree upon

Even though there exist an almost infinite number of different physical quantities, we need no more than a handful of “base” standards.



SI System of Units

Système International d'Unités (International System of Units)

A Completely Consistent Set of Basic Units• Requires NO

Conversion factors– e.g., 5280 ft = 1 mile

• Defined by UNCHANGING Physical Phenomena– Except for one...

http://www.bipm.org/en/si/



SI System History

In 1960 The 11th General Conference on Weights and Measures (GCWM) adopted the name SI System, for the recommended practical system of units of measurement.

The 1960 GCWM Specified Seven well-defined “Base” units which, by convention, are regarded as DIMENSIONALLY INDEPENDENT

http://www.bipm.org/en/si/

http://en.wikipedia.org/wiki/File:Weights_and_Measures_office.jpg



¿¿ Reader Question ??

Have you Heard of the unit:

1. METER (m)

2. KILOGRAM (kg)

3. SECOND (s)

4. AMPERE or AMP (A)

5. KELVIN (K)

6. MOLE (mol)

7. CANDELA (cd

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=7YNgY1Mh0dkOGM&tbnid=366s-nFsZqtlkM:&ved=0CAUQjRw&url=http://www.outersports.com/blog/&ei=HamgUq-sBaW1iwLUpoCACg&bvm=bv.57604443,d.cGU&psig=AFQjCNFETwwHOopuE-hB6D1DLbU1dGSJ8A&ust=1386347133423043



SI Base UnitsSI Base Units

Base quantity Name Symbol

length meter m

masskilogram

kg

time second s

electric current ampere A

thermodynamic temperature

kelvin K

amount of substance mole mol

luminous intensity candela cd

All but the kg are defined by Physical Phenomena• Examine the Defs

From this List Observe• Very common Units

– Mass (kg)– Length (m)– Time (s)

• Some Not so Common Units– Current (A)– Temperature (K)

• Some Uncommon Units– Substance amt (mol)– Luminous Int (cd)



Meter Defined

Length or Distance (meter)

Laser

1 meter

1/299792458 sphoton

“The path traveled by light in vacuum during a time interval of 1/299792458 of a second.”



kilogram Defined

Mass (kilogram)

If The ProtoType Were Cubic, its Edge Length would be About 36.2

mm (1.42”); quite small

a cylinder of PLATINUM-IRIDIUM alloy maintained under vacuum conditions by the International Bureau of Weights and Measures in Paris



Second Defined

Time (Second)

The duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom• This is the Definition of an

“Atomic” Clock– more than 200 atomic clocks are

located in metrology institutes and observatories in more than 30 countries around the world

http://www.ntp-time-server.com/images/atomic%20clock%201955%20nplcaesium.gif



Amp Defined

Electric Current (ampere)

That constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 m apart in a vacuum, would produce between these conductors a force equal to 2 x 10−7 Newton per metre of length.• What’s a Newton?→ 1kg-m/(s2)

http://www.sr.bham.ac.uk/xmm/images/fmc/wireforce_8k_398_230.jpg



Kelvin (Temperature) Defined

Thermo-dynamic temperature (Kelvin)

273.16K = 0.0098 °C Room Temperature

(72 °F) is about 295.5 Kelvins

NO “Degree” Sign Used with the Kelvin Unit

The unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.

http://www.sp.se/metrology/temperature/bilder/fasdiagram1_en.gif



mole (amt of Substance) Defined

Amount of Substance (mole)

1 mole = 6.023x1023

entities• entities must be specified

and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.

The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilograms of carbon 12.

http://www.emc.maricopa.edu/faculty/farabee/BIOBK/cisotopes.gif



Luminous Intensity Defined

Light Brightness (candela)

The are 4 (12.57) Steradians in a Sphere• 1 Str = 7.96% of the

Sphere Surface

The luminous intensity, in a given direction, of a source that emits monochromatic radiation (one-color light) of frequency 540 x 1012 Hertz (555 nm) and that has a radiant intensity in that direction of 1/683 watt per steradian

555nm

color

http://en.wikipedia.org/wiki/File:Steradian.svg



Units Have Evolved

Candela Predecessor based on a Flame• Hence the Name

Temperature Based on Freezing points• Water• Platinum

Second Based on the Sidereal (standard) day

http://zapatopi.net/lordkelvin.html



Units Have Evolved History of the Meter (or Metre)

• One ten millionth of the distance from the North pole to the equator.

• The distance between two fine lines engraved near the ends of a platinum-iridium bar

• 1 650 763.73 wavelengths of a particular orange-red light emitted by atoms of krypton-86 (86Kr).

• The length of the path traveled by light in a vacuum during a time interval of 1/299 792 458 of a second.



SI Derived Units The Seven Base Units May be

Algebraically Combined to Produce “Derived Units”• e.g.:

m/sseconds

meters timeof units

distance of units velocityof Units

Several DerivedUnits have SpecialUsefulness and arethus Given their OWN Names



Some Derived Units

Derived quantity Name Symbol

Expression in terms of

other SI units

Expressionin terms of

SI base units

plane angle radian (a) rad - m·m-1 = 1 (b)

solid angle steradian (a) sr (c) - m2·m-2 = 1 (b)

frequency hertz Hz - s-1

force newton N - m·kg·s-2

pressure, stress pascal Pa N/m2 m-1·kg·s-2

energy, work, quantity of heat

joule J N·m m2·kg·s-2

power, radiant flux

watt W J/s m2·kg·s-3

electric charge, quantity of electricity

coulomb C - s·A



Some (more) Derived UnitsDerived quantity Name Symbol

Expression in terms of

other SI units

Expressionin terms of

SI base units

electric potential difference,electromotive force

volt V W/A m2·kg·s-3·A-1

capacitance farad F C/V m-2·kg-1·s4·A2

electric resistance

ohm V/A m2·kg·s-3·A-2

electric conductance

siemens S A/V m-2·kg-1·s3·A2

magnetic flux Weber Wb V·s m2·kg·s-2·A-1

magnetic flux density

tesla T Wb/m2 kg·s-2·A-1

inductance henry H Wb/A m2·kg·s-2·A-2

Celsius temperature

degree Celsius °C - K

luminous flux lumen lm cd·sr (c) m2·m-2·cd = cd

illuminance lux lx lm/m2 m2·m-4·cd = m-

2·cd



SI prefixes – A form of ShortHand Factor Name Symbol Factor Name Symbol

1024 yotta Y 10-1 Deci d

1021 zetta Z 10-2 Centi c

1018 exa E 10-3 milli m

1015 peta P 10-6 micro µ

1012 tera T 10-9 nano n

109 giga G 10-12 pico p

106 mega M 10-15 femto f

103 kilo k 10-18 atto a

102 hecto h 10-21 zepto z

101 deka da 10-24 yocto y



Derived Units Family Tree

No

Sp

ec

ial Na

me

s



Old (and Tired) Unit Sets

MKS• Stands for Meter-Kilogram-Second as the

Most Common Units– Predecessor to The SI System

CGS• Means Centimeter-Gram-Second

– Still Widely Used

IPS, FPM, FPH• Inch-Pound-Sec, Foot-Lb-Min, Ft-Lb-Hour



American Engineering System, AES – Still in (declining) UseFundamental Dimension Base Unit

length

mass

force

time

electric charge [Q]

absolute temperature

luminous intensity

amount of substance

foot (ft)

pound (lbm)

pound (lbf)

second (sec)

coulomb (C)

degree Rankine (oR)

candela (cd)

mole (mol)

Some Are the

SAME SI



Conservation of Units

Principle of conservation of units:• Units on the LEFT side of an equation

MUST be the SAME as those on the RIGHT side of an Equation

Then Have Dimensional Homogeneity• Needed to Prevent

“Apples & Oranges” Confusion– e.g., I Buy 100 ft of Wire at One Store and 50 m

at another; how much total Wire do I have? (It’s NOT “150”)



Unit Conversion by Chain-Link

To Determine the Amount of Wire I have I Need to Convert to Consistent (Homogeneous) Units

Start by Thinking About the Definition of “1”• AnyThing divided by ITSELF = “1”• Now Consider a “minute” minute 1 Seconds60

1 mintherefore 1

60 sec

60 secor 1

1 min

Read as “60 Seconds per minute”



Chain-Link Unit Conversion

Units can also be Multiplied and Divided in a manner similar to Numbers• This how we get, say, “Square Feet”

– e.g.; Consider an 8ft x 10ft Engineer’s Cubicle in Dilbert-Land. How Much WorkSpace Does the Engineer Have?

2ft 80 ftxft8x10 ft 108ft x rkSpc W

Now Back to the Wire• Want to Know how many FEET of Wire

I have in Total



Chain-Link Unit Conversion cont.

Check in Table 16.8 and Find “3.2808 ft = 1meter (“3.2808 ft per meter”)• Multiply the 50m by this special Value of 1

feet 164.04meter 1

feet 3.2808meter 50 1meter 50

Can “Cancel” The Units by Division

So then the Total Wire = 264 ft



Chain Link Examples A World-Class Sprinter can Run 100m in 10s.

• How Fast is this in MPH?

hr

miles 37.22

hr 1

min 60

min 1

s 60

ft 5280

mile 1

m 1

ft 3.2808

s 10

m 100

Gasoline In Seoul Costs 1840 Korean-Won (W) for one Liter of Regular Unleaded• How Much is this in $ per Gallon

– Find Currency Exchange Rate → $1 = 1150 W

Gal

$ 06.6

Gal 7.48

ft 1

ft 1

Liter 28.317

W 1150

$ 1

liter 1

W 1840 3

3



Several Forms of “1”

Unit Conversion Factors1 mile = 5280 feet 1 Btu = 1054.4 Joule 1 hour = 60 min 1 meter = 3.281 feet 1 Watt = 1 Joule/sec 1 min = 60 sec 1 foot = 12 inches 1 HorsePower = 2545 Btu/hr 1 gallon = 3.785 liters 1 yard = 3 feet 1 km = 1000 meters 1 ft3 = 7.4805 gallons 1 lb = 4.448 Newtons 1 furlong = 220 yards 1 Pascal = 1 Newton/m2 1 m2 = 1973.5 Circular inch °F = 1.8x°C + 32 1 HorsePower = 550 ft-lb/s 1 cup = 48 TeaSpoon 1 Acre = 43,560 ft2 1 lb = 16 ounces $1 = 0.787 € $1 = 16,030 Viet Nam Dong $1 = 10.825 Mexican Pesos

ANYTHING Divided by ItSelf = 1

USD1

MXNs 10.825 1

Btu 1

J 1054.41

HP 1

slbsft 5501

acre 1

ft 43560 2



Units – Exponent Properties1 as an exponent a1 = a

0 as an exponent a0 = 1

Negative Exponents(flippers)

The Product Rule

The Quotient Rule

The Power Rule (am)n = amn

The Product to a Power Rule

(ab)n = anbn

The Quotient to a Power Rule

.n n

n

a a

b b

.m

m nn

aa

a

.m n m na a a

1, ,

n nn mn

n m n

a b a ba

b aa b a

This sum

mary assum

es that no denom

inators are 0 and that 00 is not

considered. For any integers m

and n



Raising units to POWERS Start again with 1

Can do the SAME Thing with Units.

And 123 = 1728 so

Thus have 1728 “cubic inches” per “Cubic Foot”

What’s a “Cubic Yard” in “Cubic Feet”?

So have 27 cubic-ft per cubic-yd• NOT “9”

n1111 32

33

3333

foot 1

inches 12

foot 1

inches 1211

3

3

33

33

foot 1

inches 1728

foot 1

inches 121

33

333

yd 1

feet 3

yd 1

feet 31



7 inches, Water Column Ms. Ezersky noted

that Natural Gas is delivered by PG&E to home at a pressure of 4-7 “inches of Water Column”

This is a unit of pressure, Just Like Pascals or psig

A U-Tube Manometer can Measure Pressure Differences in Inches of Water Column

http://images.yourdictionary.com/images/americanheritage/A4manome.jpg



7 inches, Water Column To Calc the “in-WC”

pressure we need to know some Engineering Physics

From ENGR36

• Where– ≡ Liquid SPECIFIC

WEIGHT– h ≡ Liquid Column

Height

For Liquid Water at Room Temperature and Pressure

Now find 7 in-WC in psig

hPg

3mN 9790w

Natural Gas @ 9.5 inWC



7 inches, Water Column

Convert out the N & m

33 m

inN68530in7

m

N9790

wwg hP

33

3 12in

1ft

3.281ft

1m

4.448N

lb1

m

inN68530

gP

23

3

3

3

3 in

lb2524.0

1728in

1ft

35.32ft

1m

m

inlb15407

gP



White Board Examples cont.

The USA FDA recommends that Adults consume 2200 Calories per Day

• What then is the “Power Rating” of a Grown Human Being?

– Note that there are TWO types of “Calories”

1. The Amount of Heat Required to Raise the Temperature of 1 GRAM of water by 1 °C (or 1 Kelvin) Often Called the Gram-CAL; This is what is in the Text

2. The Amount of Heat Required to Raise the Temperature of 1 KILOgram of water by 1 °C Often Called the kgCAL or kiloCal; This is what

you read on the side of Food Packaging



Tire Pressure Many AutoMobile

Tires have a Maximum Pressure Rating of About 44 psig.

Convert 44 psi to kiloPascals (kPa)



Ton of Refrigeration

During his Presentation Mr. Ian McClaren of SouthLand Industries described the “Ice Storage” Cooling System Behind Bldg-1800.

He Noted that the Cooling Power of this system was Rated in “Tons”

What is a “Ton” of Cooling Power



Ton of Refrigeration

A TON of the refrigeration is defined, roughly, as the COOLING effect of melting 2000 lbs of water ICE over a 24 HOUR Period• From PHYS4C (or ASHRAE HandBook)

find that the “Latent Heat of Fusion” for ice is 333.55 kJ/kg

On WhtBoard Convert a “Ton of Refrigeration” to • kW and Btu/hr



White Board Examples

A 2003 Chevy z06 corvette• Has a 5.7 Liter V8 Engine

– What is the Engine Displacement in cubic-inches?

• Develops 410 HP– What is the Power in Watts?

A the Maximum recommended pressure for many 65R15 tires is 44 psi (lbs per sq-inch; NOT lbs) • What is this Max Pressure in kPa?



Percent (%) – What it is?

Divide the Word into Two Parts• “PER” ≡ DIVIDE• “CENT” ≡ HUNDRED

– e.g., 1¢ (1 cent) = 1/100th of a Dollar ($)

Example – Home Prices• The median home price in the USA Was

13.6% Lower in 2009 than in 2008– Meaning for every $100k of home value in

2008, the Market prices Decreased by $13.6k in 2009



PerCent Notation

n% Means

100

n

• “n per hundred”

And

100

1

100n

n

But by Decimals

01.0100

1

So n%

nn

01.0100



PerCent as a UNIT

100% Means

1100

100

In other Words

11

%100

Convert 37.3% using 100%/1 = 1

373.0%100

1%3.37

Suggest Using Every Time

11

%100or 1

%100

1



Example – Covert to Decimals

Convert to Decimals a) 94% b) 7.6%

Solutions

94.001.094100

194

%100

1%94

076.001.06.7100

16.7

%100

1%6.7



Convert TO %-Notation

Convert a) 2.39 b)0.2 c) 3/8

Solutions Using 100%/1 = 1

%239%10039.21

%10039.2

%20%1002.01

%1002.0

375.08

3 %5.37%100375.0

1

%100375.0



PerCent Change ≡ Δ% Any Percentage

Change; the “∆%”, in any quantity is expressed Relative to a BASELINE

• Δ% can be either Positive or Negative– Positive Δ% → the

New (or Changed) value is GREATER than the BaseLine

– Negative Δ% → the New (or Changed) value is Less Than than the BaseLine

Example• My Old Car got

37mpg. My new truck gets 29mpg– Find %-Change in the

Fuel Efficiency

1

%100

Baseline

Baseline New%

1

%100

37mpg

37mpg 29mpg%

%6.211

%100

37

8%



Example % Change The Graph at right

shows the magnetic B-H behavior of an Iron-Silicon Alloy

The Graph To finch the %-Change in Flux Density when the Field Strength changes from 20 Amps per meter (A/m) to 90 A/m



Example % Change

1.22

1.601

%100

Baseline

Baseline New%

1

%100

A/m 1.22

A/m 1.22 A/m 1.60%

90

1

%100

1.22

0.38%

%1.31%

Thus B increases by 31.1% when H increases by 350%



All Done for Today

How toSpend

theCalories






Bruce Mayer, PE Chabot College MathematicsTir

e P

ress

ure

: 44

psi

→ k

Pa













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