Download Units Significant Figures (SF): Summary Dimensional Analysis

Units
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Two basic systems used by engineers
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SI
US Customary
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We need to work in both
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Significant Figures (SF): Summary
Know the fundamental and derived units in both systems.
Know how to convert within and in between both systems.
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All answers require units
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As with all rules there are usually exceptions
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Unit vectors (Chapter 3) do not have units.
Strain (Chapter 12) is considered unitless even though it is
derived from a length/length calculation.
Dimensional Analysis
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The study of the dimensions of an equation.
Can be used to check derivations.
Can be used to determine relationships between
variables.
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Full credit on exams requires appropriate use of
significant figures
Systematic Errors
v = 2 gh
g = acceleration due to gravity
h = height
v = velocity
Is this correct? Let’s check!
How precisely do you know the reported value?
Types of Errors
Example: In Chapter 2 we will derive the equation
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Use all available SF in calculations – do not round off
until the end
For the final answer, follow the given rules to
determine the SF of the final answer
Be realistic about the number of significant figures you
report
A fault in our equipment / procedure that produces the
same error every time we make a measurement.
That is, the error is repeatable (but may be unknown).
If known, the error is correctable.
Example:
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I use a “cheap” tape measure to ascertain the height of
everyone in the class.
I compare my tape measure with one at NIST (The National
Institute of Standards and Technology) and I find that my
tape measure is in error by 0.1 inches / foot to the “low” side.
I can adjust or correct everyone’s height by adding 0.1 inches
/ foot to their height.
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Types of Errors
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Accuracy vs. Precision
Accuracy, in science, engineering, industry and statistics, is the degree of
conformity of a measured/calculated quantity to its actual (true) value.
Random Errors
A fault in our equipment / procedure that produces different
errors every time we make a measurement.
That is, the error is not repeatable (and may be unknown).
Difficult to detect and correct.
Example:
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Inaccurate
Imprecise Precise
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Precision (also called reproducibility or repeatability) is the degree that
further measurements or calculations will show the same or similar results.
In measuring the height of everyone in the class, I set a piece of
cardboard on top of their head such that it was parallel to the floor
and then measure the distance from the floor to the piece of
cardboard.
Did I have the piece of cardboard “perfectly” parallel every time?
Was I consistent in which side of the piece of cardboard I measured
to?
Did I make sure that the tape measure was “perfectly” straight?
Yadda, yadda, yadda, you get the picture.
Accurate
The shots are not
clustered (not
precise) nor near
the center (not
accurate).
The shots are not
clustered (not
precise) but near
the center
(accurate).
The shots are
clustered
(precise) but not
near the center
(not accurate).
The shots are
clustered (precise)
and near the center
(accurate).
Wikipedia: http://en.wikipedia.org/wiki/Accuracy_and_precision
Central Tendency
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For a given data set, we can characterize or
measure the “center” in a variety of ways:
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Mode: the most common (frequent) value.
Median: the “middle value.” The smallest number
such that at least half the numbers in the list are no
greater than it. If the list has an odd number of
entries, the median is the middle entry in the list after
sorting the list into increasing order. If the list has an
even number of entries, the median is equal to the
sum of the two middle (after sorting) numbers divided
by two.
The Shodor Education Foundation, Inc.: http://www.shodor.org/interactivate/dictionary/m.html
Central Tendency
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For a given data set, we can characterize or
measure the “center” in a variety of ways:
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Mean (arithmetic): The sum of a list of numbers, divided
by the total number of numbers in the list. Usually referred
to as the average.
1 n
x = ∑ xi
n i =1
The Shodor Education Foundation, Inc.: http://www.shodor.org/interactivate/dictionary/m.html
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Data Variation
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Data Variation
For a given data set, we can characterize the
“variation” in the data in a variety of ways:
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Variance and Standard Deviation: measures the
“spread” in the data.
Population
Variance
Standard
Deviation
σ2 =
σ=
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Sample
1 N
2
∑ ( xi − x )
N i =1
s2 =
1 n
2
∑ ( xi − x )
n − 1 i =1
1 N
2
∑ ( xi − x )
N i =1
s=
1 n
2
∑ ( xi − x )
n − 1 i =1
For a given data set, we can characterize the
“variation” in the data in a variety of ways:
Coefficient of Variation: unit independent. It
“normalizes” the variation measurement.
Population
CV =
Wikipedia: http://en.wikipedia.org/wiki/Standard_deviation
σ
x
Sample
CV =
s
x
Wikipedia: http://en.wikipedia.org/wiki/Coefficient_of_variation
Example
Estimation
Using months as the unit of time, calculate the mean,
median, mode, standard deviation, and the coefficient of
variation of the students’ ages at your table.
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Engineers strive for a high degree of
accuracy and precision in their work.
However, we must be aware of costs in both
time and money that are required to get our
results.
Estimation is a very useful & powerful tool /
technique that can reduce these costs.
Requires the use of “reasonableness” and
“judgment.”
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Orders of Estimation
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Order of Magnitude – Guesstimate
‰ A very rough approximation that is made quickly with very little
information and very little investment of time or money.
‰ Goal is be within 1 order of magnitude of the answer.
“Back of the Envelop Calculation”
‰ Make reasonable simplifying assumptions.
‰ Use available knowledge & experience (i.e., do not conduct any
new research).
‰ Use simplified models / equations.
‰ Goal is to be within about 20% - 40% error (not always possible).
Refined Estimates
‰ Identify key variables and refine their estimated values (may
require some research). This may cost a “little” time and money.
‰ Refine the models and equations.
‰ Goal is be within about 10%-15% or less.
Estimation Example 2
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Estimate the money spent on pizzas by all
the freshmen at UT during the fall semester.
Estimation Example 1
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Estimate the fuel cost required to transport all
the EF 157 students home and back this
Labor Day weekend.
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Number of students:
Number of students that will travel:
Number of students per car:
Average distance to home:
Average gas mileage:
Average fuel cost / gallon:
Result:
At The End of Every Problem
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Are my units correct?
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Do I have the correct number of SF?
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Perform a quick dimensional analysis.
Check for unit consistency (i.e., is everything in
the right set of units?).
Check the given numbers in the problem.
Follow the rules.
Do I have the right numerical answer?
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Can I do a quick estimation?
Is my answer reasonable?
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