Transcript CH 6: Eng Measurements - Electrical and Computer Engineering

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
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
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Accuracy and Precision
Not Accurate
Not Precise
Accurate but
Not Precise
Precise but
Not Accurate
Accurate
and Precise
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
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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
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
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Use of Prefixes
Convenient
method of
representing
measurements
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
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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)
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
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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.
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
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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.
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
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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
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
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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.
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
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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.)
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
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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)
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
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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
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
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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.
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
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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
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
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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
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
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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
Engineering: Fundamentals and Problem Solving, 6e
Eide  Jenison  Northup  Mickelson
Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
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