Engineering Fundamentals and Problem Solving, 6e

Chapter 6Engineering Measurements

Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup MickelsonCopyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

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

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Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup MickelsonCopyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

Accuracy and Precision

Not Accurate Not Precise

Precise but Not Accurate

Accurate and Precise

Accurate but Not Precise

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Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup MickelsonCopyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

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

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Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup MickelsonCopyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

Use of Prefixes

Convenient method of representing measurements

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Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup MickelsonCopyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

Significant FiguresAny 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)

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Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup MickelsonCopyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

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.7

Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup MickelsonCopyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

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.

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Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup MickelsonCopyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

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 9

Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup MickelsonCopyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

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.

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Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup MickelsonCopyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

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.)

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Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup MickelsonCopyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

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)

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Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup MickelsonCopyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

Arithmetic Operations and Significant Figures

General Rule for Addition and SubtractionThe 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

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Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup MickelsonCopyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

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.

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Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup MickelsonCopyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

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

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Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup MickelsonCopyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

Categories of ErrorSystematic: 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

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Engineering: Fundamentals and Problem Solving, 6e Eide Jenison Northup MickelsonCopyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.

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

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