bmayer@chabotcollege.edu mth1516_sp14_using_units.pptx 1 bruce mayer, pe chabot college mathematics...

BMayer@ChabotCollege.edu • MTH1516_Sp14_Using_Units.pptx1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

BMayer@ChabotCollege.edu

Chabot Mathematics

Units &Conversio

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/

¿¿ 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

SI Base UnitsSI Base Units

Base quantity Name Symbol

length meter m

masskilogram

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)

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

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)

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.

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.

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

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

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-

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

ial Na

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

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

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?

miles 37.22

min 60

ft 5280

mile 1

ft 3.2808

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

$ 06.6

Gal 7.48

Liter 28.317

W 1150

liter 1

W 1840 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

MXNs 10.825 1

J 1054.41

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

.m n m na a a

n nn mn

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

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

foot 1

inches 12

foot 1

inches 1211

foot 1

inches 1728

foot 1

inches 121

feet 3

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

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

3mN 9790w

Natural Gas @ 9.5 inWC

7 inches, Water Column

Convert out the N & m

inN68530in7

wwg hP

3 12in

3.281ft

4.448N

inN68530

lb2524.0

1728in

35.32ft

inlb15407

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

• “n per hundred”

But by Decimals

01.0100

PerCent as a UNIT

100% Means

In other Words

Convert 37.3% using 100%/1 = 1

373.0%100

1%3.37

Suggest Using Every Time

%100or 1

Example – Covert to Decimals

Convert to Decimals a) 94% b) 7.6%

Solutions

94.001.094100

076.001.06.7100

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

%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

Baseline

Baseline New%

37mpg 29mpg%

%6.211

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

Baseline

Baseline New%

A/m 1.22

A/m 1.22 A/m 1.60%

%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

bmayer@chabotcollege.edu mth1516_sp14_using_units.pptx 1 bruce mayer, pe chabot college mathematics...

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