MEC 100 CHAPTER 4 (Engineering Estimation & Approximation)

7/24/2019 MEC 100 CHAPTER 4 (Engineering Estimation & Approximation)

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Engineering Estimations and

Approximations

CHAPTER 4

SITI MARDINI [email protected]

ROOM 3.!
mailto:[email protected]:[email protected]


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Chapter Outline

".# Introdu$tion

". Signi%i$ant Digits

".3 A$$ura$y and &re$ision

"." Types o% Errors and Error Ana'ysis

".( Engineering Estimations


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4.1 Introduction

Engineers are pro)'em so'*ers

Engineers design to satis%y a need and impro*e the 'i*ing

standard+ Examp'e, -ater is hot

Hot %or )ath/

Hot %or drin0ing/

Too hot1 not *ery hot1 et$/

+ Engineers ma0e measurements o% a *ast array o%physi$a' 2uantities pertaining to produ$t or system anden*ironment. S0i'' in ma0ing and interpretingmeasurements is an essentia' e'ement


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Objectives+ earn the di%%eren$es in4

A$$ura$y5pre$ision1

Random5systemati$ error1

6n$ertainty5error

+ 7ompute true1 %ra$tiona'1 and per$ent error

+6se proper num)er o% signi%i$ant %igures to report-or0


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4.1 Numbers &

Significant Digits


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Real and Integer Values

Real (exact or approximate) numbers represent continuous

quantities, e.g., length of rod, mass of rock, velocity of a

vehicle, etc. All type of numbers.

Integer (exact) numbers represent discrete quantities, e.g.,

number of marbles, number of people, number of

computers, etc.8or examp'e1 !(1 91 and :9(!areintegers, #.! and #; are not integers.

L


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Significant Digits

Significant digit or Significant figures is defined as any digitused in writing a number, except:

those zeros that are used only for location of the decimalpoint or

those zeros that do not have any nonzero digit on their left.

Significant figures are extremely important when reporting anumerical value.

he number of significant figures used indicates theconfidence !certainty" of that value.


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Significant Digits

#ow many$

%umber known to: %umber of sig. figures

&

'

(

)

etc.

& part per &*

& part per &**

& part per &***

& part per &****

etc.


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Significant Digits

A significant figure is an accurate digit although the last digit

is accepted to have some error.

+f length -./ cm

he number of significant figures does notinclude zeros

required to place decimal points.

exact


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Significant Digits

Significant digits allow us to systematically

express a degree of confidence in a number. A significant digit or figure is any digit used in a

number except: Zeros that are used to locate the decimal point, such

as:

*.* *.***( *.**'(** Zeros that do not have any nonzero digits on their

left, such as:

*. *.& *.'


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4.2 Accuracy and Precision


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Accuracy Accuracy 0nearness to the correct value.

1xample:

A chemistry instructor makes a .**2 sugar

solution. 3sing a sugar test, a team of students

analyzes the solution and reports the following

results:

Student esult A .*(2

4 ).562

7 '.5/2


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!recision Precision < repeata)i'ity o% the measurement

indi$ates scatterin the data

1xample:

A chemistry instructor makes a .**2 sugar solution.3sing a sugar assay, a team of students analyzes the

solution in triplicate and reports the following results:

Students esults

A .*(2, ).5-2, .*-2

4 ).)52, .'2, .*&2

7 '.5/2, -.5/2, 5.'(2


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!recision vs. Accuracy


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Measurements

+ Measurements $an )e reported as a *a'ue p'us orminus a num)er

+ Examp'e, 3.3=.

3.31 3.# and 3.( are a$$epta)'e = o8 -ith #>

A$$ura$y . o8


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4.4 Errors


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"rrors "rroris the differencebetween a measured or

calculated !reported" value and the true value.

1ngineers recognize that errors are present in

their professional lives and must be able to:

&. +dentify types of errors'. %umerically express the magnitude of errors

(. 8ecognize the confidence that may be

placed in a printed number


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Simple "rror Analysis Suppose a rod of unknown length is measured

with a standard meter stick.

Spend minutes as a team completing thisexercise:

9hat can be said about the length of the rod is

reported as:

- to / cm

-. to -.6 cm

-.- to -.5 cm$


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SystematicErrors

Systematic Errors 0 errors that can be

recognized to some regular outside

occurrence.

1ngineers must be aware of the presence of

systematic errors and eliminate those possible

and try to calculate and correct for thoseremaining.


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he error associated with systematic errors can

be corrected if the source and magnitude are

known.

8epeating measurements will not eliminateor reduce systematic errors.


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Example of Systematic Errors

+ Measuring #== m -ith ( m stee' tape

+ I% the tape is not exa$t'y (.===m1 there -i'' )esystemati$ errors "? times due to 4

Temperature e%%e$t,

Tension di%%eren$e -hen measuring tapes

Smoothnesso% the sur%a$e $an )e di%%erent


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Random"rrors

Accidental (Random) Errors- errors that occur in a

random nature.

he presence of accidental errors is evident by the

scatterin measured data.

+t is impossible to predict the magnitude and sign of

the accidental error present in any one measurement.

8epeating measurements and averaging the results

will reduce the random error in the average.


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+ Re%inement o% the apparatus and $are inits use $an redu$e the magnitude o% error

+A-areness o% the pro)'em1 0no-'edge o%the degree o% pre$ision o% the e2uipment1s0i'' -ith measurement pro$edures1 andpro%i$ien$y in the use o% statisti$s a''o- usto estimate the magnitude o% error


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Definitions here are three ways to numerically describe

error:

#. $rue error 8eported value 0 rue value

%. &ractional error rue error rue value

'. !ercent error ;ractional error < &**2


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"xample:7alculate the true error, fractional error and

percent errorfor:

eported (alue

Distance

feet

)orrect (alue

Distance

feet

&*5 &**

&** &***

) *

&- '*


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4.5 Approximations


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Approximation+ Engineers stri*e %or high


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The a$$ura$y o% these estimates depends on4

#. 7onse2uen$es5$ost o% error1. A*ai'a)'e re%eren$e materia's1

3. Time a''otted %or estimate1

". Experien$e -ith simi'ar pro)'ems.

Approximation


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+Estimate the num)er o% paper$'ips that -i'' %it in a )ox #!$m )y#=$m )y #$m

# paper $'ip #.($m )y #$m )y =.$mB

# minutes so'ution

Example

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