Engineering Estimations and ApproximationsCHAPTER 4

4.1 IntroductionEngineers are problem solversEngineers design to satisfy a need and improve the living standardLord Kelvin stated that knowledge and understanding are not of high quality unless the information can be expressed in numbers

Introduction; contExample; water is hotHot for bath?Hot for drinking?Too hot, not very hot, etc?Engineers make measurements of a vast array of physical quantities pertaining to product or system and environment. Skill in making and interpreting measurements is an essential element

Learning ObjectivesLearn the differences in: Accuracy/precision, Random/systematic error, Uncertainty/errorCompute true, fractional, and percent errorUse proper number of significant figures to report work

4.2 Numbers and Significant Digits

Real (exact or approximate) numbers represent continuous quantities, e.g., length of rod, mass of rock, velocity of a vehicle, etc.

Integer (exact) numbers represent discrete quantities, e.g., number of marbles, number of people, number of computers, etc.LInteger and Real Values

Significant DigitsSignificant digit or Significant figures is defined as any digit used in writing a number, except those zeros that are used only for location of the decimal point or those zeros that do not have any nonzero digit on their left.

Significant figures are extremely important when reporting a numerical value.

The number of significant figures used indicates the confidence (certainty) of that value.

Significant DigitsHow many?Number known to:Number of sig. figures1234etc.1 part per 101 part per 1001 part per 10001 part per 10000etc.

Significant DigitsA significant figure is an accurate digit although the last digit is accepted to have some error.

If length = 7.58 cm

The number of significant figures does not include zeros required to place decimal points.Slight errorexactexact

Significant DigitsSignificant 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:0.050.00030.002300Zeros that do not have any nonzero digits on their left, such as:0.50.5150.25

Significant DigitsDo the numbers 5000 and 5000. imply the same significance?5000. contains four significant digits.5000 is an ambiguous number. It contains either one, two, three, or four significant digits.How do you write 5000 to two significant digits?Use scientific notation: 5.0 X 103

Significant DigitsHow many significant figures should you use?The number of significant digits used implies a certain maximum error range.

Significant DigitsExample:The number 101 has three significant figures and means a number between 100.5 and 101.5. Theerror range is 1 ( 0.5) or about 1% of 101.Three significant figures implies a maximum error range of 1%.Four significant figures implies a maximum error range of 0.1%.Only in exceptional cases will precision better that 0.1% (four significant figures) be necessary in engineering problems.

Rules for Significant Digits In multiplication and division - use as many significant digits as the number that has the fewest (excluding exact conversion factors)(4.00 kg) (4 m/s2) = 16 kg m/s2(2.43)*(17.675)= ? 42.95025 Ans. 43.0 @ 4.30x101(2.479 h) (60 min/h) =? 148.74 min Ans. 148.7 @ 1.487x10Exact conversion factor

Rules for Significant DigitsIn multiplication and division(4.00x102) (2.2046 lbm/kg) =? 881.84lb Ans. 882 lbConversion factor is not exact; cannot increase precisionUse one or more significant figure for your conversion factor

Rules for Significant DigitsIn addition and subtraction - line up the decimals and retain the least significant place.897.0- 0.0922896.9078896.9 (Answer)

Rules for Significant DigitsCombined operations:If products or quotients are to be added orsubtracted, perform the multiplication anddivision first, establish the correct number ofsignificant figures in the sub answer, performthe addition and subtraction, then round tothe proper number of significant figures.

Rules for Significant DigitsCombined operations:When using calculator, it is normal practice to perform entire calculation and then report a reasonable number of significant figuresNote; 39.7/(772.3-772.26)=992.5But if 772.3-772.26=0, then it becomes impossibleUse common sense

Rules for Significant DigitsRounding827.48 rounds to 827.5 or 82723.650 rounds to 23.7 (3 significant figures)0.0143 rounds to 0.014 (2 significant figures)

Rules for Significant DigitsCombined operations:If using a calculator or computer, perform the entire operation and then round to the correct number of significant digits. Sometimes, common sense and good judgment is the only applicable rule!

Team Work1. How many significant digits are contained in each of the following quantitiesA) 5 760 000B) 222.230C) 4.626 7x102D) 0.000 6B) 1.320x103

Team WorkPerform the following computational and report with the answer rounded to the proper number of significant digits. (No numbers are exact conversions) A) 3.735-1.43B) 6.231 827x(4.23x107)C) 4500.3+372D) 4 300 240/784

Team WorkPerform the suggested calculations using exact conversions or with enough significant digits so that it does not affect the accuracy of the answer4376 ft to miles (1 mi =5 280 ft)653.545 kg to N (g=9.806 65 m/s2)7.8*1010 atoms to mole (NA=6.022 136 736*1023 atoms/mol)

4.3 Accuracy and Precision

AccuracyAccuracy - nearness to the correct value.

Example:A chemistry instructor makes a 5.00% sugar solution. Using a sugar assay, a team of students analyzes the solution and reports the following results: Student Result A 5.03% B 4.96% C 2.98%

PrecisionPrecision - repeatability of the measurement indicates scatter in the dataExample:A chemistry instructor makes a 5.00% sugar solution. Using a sugar assay, a team of students analyzes the solution in triplicate and reports the following results: Student Result A 5.03%, 4.97%, 5.07% B 4.49%, 5.52%, 5.01% C 2.98%, 7.98%, 9.23%

Precision vs. Accuracy

MeasurementsMeasurements can be reported as a value plus or minus a numberExample; 32.30.232.3, 32.1 and 32.5 are acceptableExample 220 oF with 1% accuracy2.2 oF

4.4 Errors

ErrorError is the difference between a measured or calculated (reported) value and the true value.Engineers recognize that errors are present in their professional lives and must be able to:1. Identify types of errors2. Numerically express the magnitude of errors3. Recognize the confidence that may be placed in a printed number

Simple Error AnalysisSuppose a rod of unknown length is measured with a standard meter stick.Spend 5 minutes as a team completing this exercise: What can be said about the length of the rod is reported as:7 to 8 cm7.5 to 7.6 cm7.57 to 7.59 cm?

Systematic ErrorsSystematic Errors - errors that can be attributed to some regular outside occurrence.

Engineers must be aware of the presence of systematic errors and eliminate those possible and try to quantify and correct for those remaining.

Systematic ErrorsThe error associated with systematic errors can be corrected if the source and magnitude are known.

Repeating measurements will not eliminate or reduce systematic errors.

Example of Systematic ErrorsMeasuring 1200 m with 25 m steel tapeIf the tape is not exactly 25.000m, there will be systematic errors 48 timesTemperature effect; Can be corrected by using thermal expansion coefficient But again, thermometer error

Example of Systematic ErrorsTension difference when measuring tapesSmoothness of the surface can be different

Random ErrorsAccidental (Random) Errors - errors that occur in a random nature.

The presence of accidental errors is evident by the scatter in measured data.

Random ErrorsIt is impossible to predict the magnitude and sign of the accidental error present in any one measurement.

Repeating measurements and averaging the results will reduce the random error in the average.

Example of Random ErrorsReading graduation levelThe measuring tape may sag during measurementTo correct this error, calculate deflection using A, I, E, tension T; all involves errors

Random ErrorRefinement of the apparatus and care in its use can reduce the magnitude of errorAwareness of the problem, knowledge of the degree of precision of the equipment, skill with measurement procedures, and proficiency in the use of statistics allow us to estimate the magnitude of error

What type of error is it?The produce scale at the grocery store has water on it. (The water runs off the produce)The timekeeper sneezes at the moment the runners cross the finish line.Gasoline sloshed from your tank prior to the pump shutting off (mpg calculation).Measuring with a 100 ft tape that is actually 99.01 ft.Press the wrong key(s) on a calculator during a long calculation.

DefinitionsThere are three ways to numerically describe error:1) True error = Reported value - True value2) Fractional error = True error / True value3) Percent error = Fractional error * 100%

An example:Calculate the true error, fractional error andpercent error for:Reported ValueDistancefeetCorrect ValueDistancefeet1091001050100045501720

4.5 Approximations

ApproximationEngineers strive for high-level precisionAlso, it is important to be aware of an acceptable precision and the time and cost of attaining itEngineers are expected to make an approximation to the solution before time and funds invested to increase accuracy

Approximation; contEngineers rely on their basic understanding of the problem under discussion coupled with their previous experience

Approximation; contThe accuracy of these estimates depends on:

1. Consequences of error,2. Available reference materials,3. Time allotted for estimate,4. Experience with similar problems.

Example of ApproximationA city with 12000 population tries to solve solid waste problem for next ten years. A city council asks a civil engineer how many acres of land will be needed for the disposal of solid waste.

Example of ApproximationA civil engineer quickly surveyed that the national average of the solid waste is 2.75 kg/capita/day(2.75 kg/capita/day)*(365 days/year)Approximately 1000 kg/year/personMaybe compacted to 400 to 600 kg/m3

Example of ApproximationApproximately the per capita landfill volume is 2 m3 each yearAbout the refuse of 2000 people per year needs one acre if filled 1 m deep (1 acre =4 047 m2)The city bedrock is 6 m deep, so 4 m deep fill is possible1.5 acres per year, 15 acres for 10 yearsGrowth factor of the city, recommend 20 acres

Team WorkEstimate the number of paper clips that will fit in a box 16 by 10 by 12 in

*Bring to class (suggest print handouts 6/page then copy 1-2 sided to minimize your load; each handout has 10 or 11 slides):significant digits handout (1 per team) error handout (1 per team)*