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Engineering Fundamentals and Problem Solving, 6e

Engineering Fundamentals and Problem Solving, 6e. Chapter 6 Engineering Measurements and Estimations. Chapter Objectives. Determine the number of significant digits in a measurement

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Engineering Fundamentals and Problem Solving, 6e

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  1. Engineering Fundamentals and Problem Solving, 6e Chapter 6 Engineering Measurements and Estimations

  2. Chapter Objectives • Determine the number of significant digits in a measurement • Perform numerical computations with measured quantities and express the answer with the appropriate number of significant digits • Define accuracy and precision in measurements • Define systematic and random errors and explain how they occur in measurements

  3. Chapter Objectives (Cont’d) • Solve problems involving estimations of the required data and assumptions to enable a solution • Develop and present problem solutions, involving finding or estimating the necessary data, that enable others to understand your method of solution and to determine the validity of the numerical work

  4. Accuracy and Precision Not Accurate Not Precise Accurate but Not Precise Precise but Not Accurate Accurate and Precise

  5. Presentation of Numbers • Less than zero: 0.234 not .234 • Divide numbers of three orders of magnitude or more with spaces not commas: 1 234.432 1 not 1,234.432,1 • Use scientific notation for compactness: 9.87(10)6 not 9 870 000

  6. Use of Prefixes Convenient method of representing measurements

  7. Any digit used to express a number, except those zeros used to locate the decimal point. Examples: 0.00123 (3 significant figures) 1.00123 (6 significant figures) 1 000 000 (1 significant figure) 1.000 000 (7 significant figures) 0.100 (3 significant figures) Significant Figures

  8. Significant Figures Use scientific notation to clarify significant figures Example: 3 000 (1, 2, 3, or 4 sig. fig?) 3(103) (1 significant figure) 3.0(103) (2 significant figures) etc.

  9. Measurements • Counts (exact values): All digits are significant 32 baseballs (2 sig. fig.) 5 280 ft in a mile (4 sig. fig.) • Measured Quantities • Measurements are estimates. The number of significant figures depends upon several variables: • instrument graduations, • environment, • reader interpretation, etc.

  10. Measurements (con’t) • Bar is between 2 and 3 inches • Think of it as 2.5 ± 0.5 inches • Estimate between 2.6 and 2.7 inches or 2.65 ± 0.05 inches • “Best” estimate 2.64 inches with the understanding that • the 4 is doubtful

  11. Measurements (con’t) Standard practice: In a measurement, count one doubtful digit as significant. Therefore the length of the bar is recorded as 2.64. For calculation purposes the result has 3 significant figures.

  12. Arithmetic Operations and Significant Figures General Rule for Rounding To round a value to a specified number of significant figures, increase the last digit retained by 1 if the first figure dropped is 5 or greater. 15.750 becomes 15.8 (3 sig. fig.) 0.015 4 becomes 0.15 (2 sig.fig.) 34.49 becomes 34.5 (3 sig. fig.) or 34 (2 sig. fig.)

  13. Arithmetic Operations and Significant Figures General Rule for Multiplication and Division The product or quotient should contain the same number of significant digits as are contained in the number with the fewest significant digits. Examples (15)(233) = 3495 (4 sig. fig. if exact numbers) (15)(233) = 3500 (2 sig. fig. if numbers are measurements) (24 hr/day)(34.33 days) = 823.9 hr (4 sig. fig.) (since 24 is an exact value)

  14. Arithmetic Operations and Significant Figures General Rule for Addition and Subtraction The answer should show significant digits only as far to the right as seen in the least precise number in the calculation. Note: last digit in a measurement is doubtful. Example (color indicates doubtful digit) 237.62 28.3 119.743 385.663 By our rules, we keep one doubtful digit. The answer is 385.7

  15. Arithmetic Operations and Significant Figures Combined Operations • With a calculator or computer, perform the entire calculation and then report result to a reasonable number of significant figures. • Common sense application of the rules is necessary to avoid problems.

  16. Accounting for Errors in Measurements Measurements can be expressed in 2 parts: • A number representing a mean value of the physical quantity measured • An amount of doubt (error) in the mean value Example 1: 52.5 ± 0.5 Example 2: 150 ± 2% so 150 means: 147 - 153 The amount of doubt provides the accuracy of the measurement

  17. Categories of Error Systematic: Error is consistently in the same direction from the true value. - Errors of instrument calibration - Improper use of measurement device - External effects (e.g. temperature) on measurement device - Must be quantified as much as possible for computation purposes

  18. Categories of Error (con’t) Random: Errors fluctuate from one measurement to another for the same instrument. - Measurements usually distributed around the true value - May be caused by sensitivity of instrument - Statistical analysis required

  19. Engineering Estimations Variables influencing an estimation • Time/Money available • Sources of information available • Computational equipment available • Presentation requirements - Verbal to supervisor/colleagues - Internal company presentation - Public forum

  20. Components of an Estimation Presentation • Clear statement of problem • List of assumptions • Data available • Appropriate computations, sufficient for others to evaluate your work • Conclusions, discussion and recommendations

  21. Estimation Example Example Problem 6.4 Estimate the cost of concrete that is in the roadbed of Interstate 17 running from Flagstaff, Arizona, to Interstate 10 near the Phoenix airport. Keep track of the time to develop a solution and perform the write-up.

  22. Example –cont’d • ASSUMPTIONS • Four lanes (two lanes each direction) from Flagstaff to Loop 101 in Phoenix. • Eight lanes in Phoenix • Assume entire roadbed is concrete • Neglect on and off ramps, bridge supports and railings, and emergency stop lanes.

  23. Example –cont’d • COLLECTED DATA • Concrete cost $95 per cubic yard delivered to site (estimate from contractor) • Average depth of roadbed is 12 inches (Department of Transportation) • Lane width is 12 feet (Department of Transportation) • Unit relationships: 1 mi = 5280 ft • 1 yd = 3 ft • 1 ft = 12 in • Data from 2009 State Farm Road Atlas • miles lanes • Flagstaff to Phoenix Loop 101 124 4 • Loop 101 21 8

  24. Example –cont’d CALCULATIONS Volume (V) = Length (L) x Width (W) x Depth (D) Cost (C) = Volume (V) x (cost/cubic yard) Va = (124 mi)(5280 ft/mi)(4 lanes)(12 ft)(12 in)(1 ft/12 in)(1 yd3/27 ft3) = 1.164(10)6 yd3 Vb = (21 mi)(5280 ft/mi)(8 lanes)(12 ft)(12 in)(1 ft/12 in)(1 yd3/27 ft3) = 0.3942(10)6 yd3 V = Va + Vb = 1.558(10)6 yd3 C = (1.588(10)6 yd3)($95/yd3) = $148,000,000

  25. Example –cont’d • Discussion/Conclusions • Cost of concrete may vary considerably in a short time period. $95 cost per yard used was an average of several reports found on Internet. • Time estimate: 40 min (obtaining data and calculations) + 30 min (write-up) = 70 min

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