Units Estimation

8/12/2019 Units Estimation

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Units & Estimation

Freshman Clinic I


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Units

Physical Quantities

Dimensions

Units


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Physical Quantities

Measurement of physical quantities, e.g.,length, time, temperature, force

To specify a physical quantity, comparemeasured numerical value to a referencequantity called a unit

A measurement is a comparison of howmany units constitute a physical quantity


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Physical Quantities

If we measure length (L) and use meters as units,

and L is 20 of these meter units, we say that

L=20.0 meters (m) For this relationship to be valid, an exact copy of

the unit must be available, i.e., a standard

Standards: set of fundamental unit quantities kept under

normalized conditions to preserve their values asaccurately as possible


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Dimensions

Used to derive physical quantities

NOTE: Dimensions are independent of units;

for a given dimension there may be manydifferent units

Length is represented by the dimension L

Others physical quantities are time T, force

F, mass m


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Kinds of Dimensions

Fundamental dimensioncan beconveniently and usefully manipulated

when expressing physical quantities for aparticular field of science or engineering

More simply, a basic dimension

Velocity, e.g., can be considered afundamental dimension but we customarilytreat it as a derived dimension (V=L/T)


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Units

Each fundamental dimension requires a

base unit

BUT (!), there are many unit systems thatcan be used with a given dimension system


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Units

The International System of Units (SI)

serves as an international standard to

provide worldwide consistency Two fundamental unit systems exist today

the meter-kilogram-second (MKS) used

worldwide and the Engineering Systemfoot, pound force, second used in the US


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SI Units

Seven base units are defined so that they can bereproduced

Length meter mTime second s

Mass kilogram kg

Electric current ampere A

Temperature kelvin K

Amount of substance mole mol

Luminous intensity candela cd


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SI Units

Table 6.4 lists derived units with special names

Table 6.5 lists derived units that are combinations

of units with special names and base units Unit Prefixes are listed in Table 6.6. They can be

used to eliminate non-significant zeros and leading

zeros

It is customary to express a numerical value as a

number between 0.1 and 1000 with a prefix


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More About Prefixes

Use prefixes or scientific notation to

indicate significance

10.000 km 5SF 9999.5-10000.5 m

10.00 km 4SF 9995-10005 m

10.0 km 3 SF 9950-10050 m

10 km 2 SF 5000-15000 m


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Rules for SI Units

Periods not used

Lower case unless derived from proper

name

Do not add s to pluralize symbols

Leave a space between numerical value and

symbol (except degrees, minutes, and

seconds of angles and degrees Celsius)


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More Rules

Plurals of the unit name (not the symbol) are

formed as necessary except for lux, hertz, and

siemens No hyphens or spaces between prefix and unit

name

Omit final vowel in megohm, kilohm, and hectare

Use symbols with numerical values; use names

with numerical value written in words


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Multiplication/Division

For unit products leave one space betweenunits or use a hyphen. For symbol products

use a center dot. Use the word per in a quotient; use the

slash (/) with symbols or unit-1

For powers use squared or cubed afterthe unit name. For area or volume, placethe modifier before the unit name.


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US Customary System

Quantity Unit Symbol

Mass slug slug

Length foot ft

Time second s

Force pound lb


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US Engineering System

Quantity Unit Symbol

Mass pound mass lbm

Length foot ft

Time second s

Force pound force lbf


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Conversion of Units

Dimensional Analysis

1 meter = 3.2808 feet x 1 minute = 0.05468 feet

minute meter 60 seconds second


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Estimation

Significant Digits (Significant Figures)

Accuracy and Precision

Approximations


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Significant Digits(www.batesville.k12.in.us/Physics)

All non-zero digits are significant digits.

4 has one significant digit

1.3 has two significant digits

4,325.334 has seven significant digits


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Significant Digits

(www.batesville.k12.in.us/Physics)

Zeros that occur between significant

digits are significant digits.

109 has three significant digits

3.005 has four significant digits

40.001 has five significant digits


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Significant Digits


Zeros to the right of the decimal point and to

the right of a non-zero digit are significant. 0.10 has two significant digits

leading zero is not significant, but the trailing zero is significant)

0.0010 has two significant digits (the last two)

3.20 has three significant digits

320 has two significant digits

zero is to the left of the decimal point - not significant.)

14.3000 has six significant digits

400.00 has five significant digits

two zeros to the right of the decimal point are significant because they are to the right of

the "4". The two zeros to the left of the decimal point are significant because they lie

between significant digits.


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Significant Digits


These three rules have the effect that all

digits of the mantissa (number part) are

always significant in a number written inscientific notation.

2.00 x 107has three significant digits

1.500 x 10-2has four significant digits


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Multiplication and Division

Answer should have same number ofsignificant digits as in number with fewest

significant digits. e.g., (2.43)(17.675)=42.95025 should be

expressed as 43.0 (3 significant digits, sameas 2.43, not 7-the actual product)


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More Examples

Using an exact conversion factor

(2.479 hr)(60 min/hr)=148.74 minutes (5SF?)

Express the answer as 148.7 minutes (4SF, same asin the number 2.479)

Conversion factor not exact

(4.00x102

kg)(2.2046lbm/kg)=881.84 lbm (5SF?)Express the answer as 882 lbm (3 SF as in 4.00x102

kg)


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One More

Quotient

589.62/1.246=473.21027 (Should this be 8

SF?)

Express the answer as 473.2 which is correct

to 4SF, the number of SF in 1.246)


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Addition and Subtraction

Show significant digits only as far to theright as is seen in the least precise number

in the calculation (the last number may bean estimate).

1725.463

189.2 (least precise)

16.73

1931.393 Report as 1931.4


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More on Addition and

Subtraction897.0


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Combined Operations

When adding products or quotients, perform the

multiplication/division first, establish the correct

number of significant figures, and thenadd/subtract and round properly.

If results of additions/subtractions are to be

multiplied/divided, determine significant figures

as operations are performed. If using a calculator,report a reasonable number of significant figures.


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Rules for Rounding

Increase the last digit by 1 if the first digit

dropped is 5 or greater

827.48 becomes 827.5 for 4 SF

827.48 becomes 827 for 3 SF




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Accuracy is the measure of the nearness of a given

value to the correct or true value.

Precision is the repeatability of a measurement,i.e., how close successive measurements are to

each other.

Accuracy can be expressed as a range of values

around the true value, usually shown as a valuewith a +/- range. 32.3+0.2 means that the true

value lies between 32.1 and 32.5


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The range of a permissible error can also be

expressed as a percentage of the value.

Consider a thermometer where the accuracy isgiven as + 1% of full scale. If the full scale

reading is 220oF then readings should be

within + 2.2o of the true value, i.e.,220x0.01=2.2


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Approximations

Precision is a desirable attribute of

engineering work

You do not always have time to be precise

You need to be able to estimate

(approximate) an answer to a given problem

within tight time and cost constraints.


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Approximations

A civil engineer is asked to estimate the

amount of land required for a landfill. This

landfill will need to operate for the comingten years for a city of 12000 people.

How would you approach this estimation

problem?


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Approximations

The engineer knows that the nationalaverage solid waste production is 2.75 kg

per person per day. He then estimates thateach person will generate

(2.75 kg/day)(365 days/year) = 1000 kg/year

The engineers experience with landfillssays that refuse can be compacted to 400-600 kg/m3.


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Approximations

This leads to the conclusion that the perperson landfill volume will be 2 m3per

year. One acre filled 1 m deep will hold one

years refuse of 2000 people. (We get thisfrom 1 acre =4047 m2).

The area requirement would then be 1 acrefilled to a depth of 6 meters.


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Approximations

But the engineer knows that bedrock exists

at the proposed site at a depth of 6 feet. So

the estimated depth needs to be reduced to 4feet and the area needs to be increased to

1.5 acres for 1 year, or 15 acres for a 10

year landfill life.


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Approximations

To allow for expected population growth

the engineer revises the final estimate to 20

acres for a landfill life of 10 years.


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Now Its Your Turn

Estimate the cost to launch a

communications satellite. The satellite

should have a life of 12 years. The satellite has 24 transponders plus 6

spares that weigh 12 pounds each.


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Communications Satellite

Each transponder requires:

20 lbs. of avionics

40 lbs. of batteries and solar cells

The satellite uses 80 pounds of station-

keeping fuel per year

The satellite carries an apogee kick motor that

weighs 3000 lbs.


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Launch Vehicle

Cost to launch on a Delta rocket is

$8000/lb. per lb. up to 6000 lb. and

$10000/lb. for each pound over 6000 lbs. Cost to launch on an Atlas-Centaur rocket is

$9000 per lb.

Which is the more economical launchvehicle for this spacecraft?


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Solution

24 transponders plus 6 spares at 12 lbs. eachweighs 360 lbs.

20 lbs. of avionics per transponder (30)weighs 600 lbs.

40 lbs. of batteries and solar cells pertransponder (30) weighs 1200 lbs.

80 lbs. of station-keeping fuel per year (12)weighs 960 lbs.


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Spacecraft Total Weight

Transponders 360 lbs.

Avionics 600 lbs.

Batteries and solar cells 1200 lbs

Station -keeping fuel 960 lbs.

Spacecraft weight 3120 lbs.

Apogee kick motor 3000 lbs.

Total weight at launch 6120 lbs.


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Launch Costs

For Delta: (6000 lbs.)($8000)

+(120 lbs.)($10000) = $49.2M

For Atlas-Centaur: (6120 lbs.)($9000) =$55.08M

Launching on Delta is cheaper by $5.88M

Units Estimation

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