Engineering Fundamentals and Problem Solving, 6e Chapter 6 Engineering Measurements
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 2
Accuracy and Precision Not Accurate Not Precise Accurate but Not Precise Precise but Not Accurate Accurate and Precise 3
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 4
Use of Prefixes Convenient method of representing measurements 5
Significant Figures 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) 7
Significant Figures Use scientific notation to clarify significant figures Example: 3 000 (1, 2, 3, or 4 sig. fig?) 3(10 3 ) (1 significant figure) 3.0(10 3 ) (2 significant figures) etc. 8
Measurements Counts (exact values): All digits are significant Measured Quantities 32 baseballs (2 sig. fig.) 5 280 ft in a mile (4 sig. fig.) Measurements are estimates. The number of significant figures depends upon several variables: instrument graduations, environment, reader interpretation, etc. 9
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 10
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. 11
Arithmetic Operations and General Rule for Rounding Significant Figures 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.) 12
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) 13
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 14
Arithmetic Operations and Combined Operations Significant Figures 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. 15
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 16
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 17
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 18