Post on 02-Jan-2016
description
Unit Analysis
“Measurement units can be manipulated in a similar way to variables in algebraic relations.”
SWTJC STEM – ENGR 1201
Content Goal 15
Unit Analysis
The basis for this analysis is embodied in three rules:
1. Dimensional Consistency Rule2. Algebraic Manipulation Rule3. Transcendental Function Rule
Rule #1 – Dimensional Consistency Rule
In a unit relation, all terms must be dimensionally consistent. This means that each term must have the same units or be reducible to the same units.
SWTJC STEM – ENGR 1201
Content Goal 15
Rule 1 Dimensional Consistency
Dimension refers to “what is being measured”. For instance, when measuring the length of a table, “length” is the dimension. The unit could be feet, meters, or a variety of other “length” units.
Reducible refers to rewriting the units in fundamental units. Units are either fundamental or derived. Derived units are combinations of eight fundamental units.
Refer to Derived Units Charts on “Useful Links”
The Dimensional Consistency Rule simply reinforces the common sense idea that you can only add and subtract identical things. You cannot mix apples and oranges!
SWTJC STEM – ENGR 1201
Content Goal 15
Adding Apples and Oranges
?
SWTJC STEM – ENGR 1201
Content Goal 15
Applying Consistency Rule
x = a + b - c (Terms are separated by addition or subtraction)
terms
Consistency means terms “x”, “a”, “b”, and “c” must have the same dimension, i.e. units, in this case length/meters. If “x” is length (meters), all other terms must be length (meters)!
(meters) = (meters) + (meters) – (meters)
(length) = (length) + (length) – (length)
Example
Note that (meters) - (meters) = (meters) not zero!16 meters - 12 meters = 4 meters!
SWTJC STEM – ENGR 1201
Content Goal 15
Rule 2 Algebraic Manipulation
Rule #2 – Algebraic Manipulation Rule
Unit relations that are multiplied and/or divided can be treated like variables; i.e., canceled, raised to powers, etc.
During algebraic manipulation of a relation, dimensional consistency must be maintained. When finished, if dimensional inconsistency is noted, then either an algebraic manipulation error occurred or the original formulation of the relation was faulty.
Working through the units is a great way to check your algebra!
SWTJC STEM – ENGR 1201
Content Goal 15
Distance Formula Example
Distance formula: d = v0 . t + (1/2) . a . twhere d (m), v0 (m/s), t (s), and a (m/s2)m = (m/s) . s + (none) . (m/s2) . sm = m + m/sA problem? The relation is inconsistent!Formula is incorrect!
Distance formula: d = v0 . t + (1/2) . a . t2
m = (m/s) . s + (none) . (m/s2) . s2
m = m + mThe formula is consistent!
Examplesdimensionless constant
(no units!)
SWTJC STEM – ENGR 1201
DimAnalysis cg13d
Particle Energy Example
2
2
Particle Energy Example
1E mgh mv
2m m
where E [J], m [kg], g [ ], h [m], v [ ]ss
Assume the relation is consistant.
SWTJC STEM – ENGR 1201
DimAnalysis cg13d
Particle Energy Example
2
2
2
Solve for v.
1mv E mgh
2
mv 2E 2mgh
2E 2mgh Ev 2( gh)
m m
Ev 2( gh)
m
SWTJC STEM – ENGR 1201
DimAnalysis cg13d
Particle Energy Example
2
Check for consistency.
Ev 2( gh)
m
kgm Nm m
ms kg s
2
mm
skg
2
2
2 2 2
2 2 2
m
s
m m m m m
s ss Consistent
s s
Rule #3 – Transcendental Function Rule
Transcendental functions (trig, exponential, etc.) and their arguments cannot have dimensions (units).
Examples of transcendental functions includes:
sin(x), cos(x), tan(x), arcsin(x)
ex
log(x), ln(x)
SWTJC STEM – ENGR 1201
Content Goal 15
Rule 3 Transcendental Function
SWTJC STEM – ENGR 1201
Content Goal 15
Transcendental Function Examples
Consider the relation A = sin(a t + b).
Neither (a t + b) nor A can have a unit.
Note that a and t can have units provided they cancel. Variable b cannot!
Suppose a = 6 Hz, t = 10 s, and b = 5 (no units). Hz is the derived unit Hertz and is reducible to fundamental units 1/s.
Then A = sin(6 1/s 10 s + 5) = sin(60 + 5) = sin(65) = 0.906
No units!
SWTJC STEM – ENGR 1201
Content Goal 15
Richter Scale Example
Earthquake intensity is measured on the Richter Scale.
MR = log(A) where is a seismic amplitude factor.
The famous San Francisco earthquake of 1906 was MR = 7.8 on the Richter scale.
Does A have units?
No! According to Rule 3, Transcendental Function
Does MR have units?
No! Ditto.
SWTJC STEM – ENGR 1201
Content Goal 15
Bernoulli Example
2
3 2
23 3 2
Bornoulli's Law
1P v g h constant
2where
kg m mP [Pa], [ ], v [ ], g [ ], h [m]
sm s1
Note: is a dimensionless constant.2
kg m kg mPa ( ) m constant
sm m s
SWTJC STEM – ENGR 1201
Content Goal 15
Bernoulli Example
2 3
N kg
m m
2
1
m
2 3
kg
s m
1
m
2m
s constant
kg m
2
2s
m 1 2 2
2 2 2
kg kgconstant
m s m skg kg kg
cons Consistant m s m s m s
tent
SWTJC STEM – ENGR 1201
Content Goal 15
Coherent Systems of Units
A system of units is coherent if all units use the numerical factor of one. For example, the SI system is coherent, so the unit m/s implies (1 meter) / (1 second).
Both the SI and USCS systems are coherent.
This means that when you use SI or USCS units in a relation (formula), no numeric factors will be needed.
Unless otherwise indicated, change all units to SI and USCS base or derived units before plugging in a formula.
SWTJC STEM – ENGR 1201
DimAnalysis cg13a
Base Units SI (Metric)
SI - Systeme International or Metric System
Fundamental Dimension
1. Length
2. Mass
3. Time
4. Temperature
5. Electric current
6. Molecular substance
7. Luminous intensity
Base Unit
meter (m)
kilogram (kg)
second (s)
kelvin (K)
ampere (A)
mole (mol)
candela (cd)
Note: Force and charge are not fundamental units.
SWTJC STEM – ENGR 1201
Content Goal 15
Derived Units in SI
SWTJC STEM – ENGR 1201
DimAnalysis cg13a
Base Units USCS
USCS - United States Customary System
Fundamental Dimension
1. Length
2. Force
3. Time
4. Temperature
5. Electric current
6. Molecular substance
7. Luminous intensity
Base Unit
foot (ft)
pound (lb)
second (s)
rankine (R)
ampere (A)
mole (mol)
candela (cd)
Note: Mass and charge are not fundamental dimensions.
SWTJC STEM – ENGR 1201
Content Goal 15
Derived Units in USCS
What is the kinetic energy of a 20 ton ship moving 5 knots?
SWTJC STEM – ENGR 1201
Content Goal 15
Coherent System Example
Ke = W · v2 / (2 · g) where Ke (lb·ft), W (lb), v (ft/s), g (32.2 ft/s2)
What is the system of units?
Related to USCS: 1 ton = 2000 lbs, 1 knot = 1.688 ft/s
Is USCS coherent? Yes.
What is the relation for kinetic energy?
Related to USCS: 1 ton = 2000 lbs, 1 knot = 1.688 ft/s
What is first step?
SWTJC STEM – ENGR 1201
Content Goal 15
Coherent System Example
Ke = W · v2 / (2 · g) where Ke (lb·ft), W (lb), v (ft/s), g (32.2 ft/s2)
Convert to base units.
W in tons and v in knots.
20 tons · 2000 lbs/ton = 40,000 lbs = 4 · 104 lbs
What’s not in base units?
5 knots · 1.688 (ft/s)/knot = 8.44 ft/s
Substituting,
Ke = 4 · 104 lb · (8.44 ft/s) 2 / (2 · 32.2 ft/s2)
SWTJC STEM – ENGR 1201
Content Goal 15
Coherent System Example
Ke = 4 · 104 · 71.2336 / 64.4 lb · ft 2 /s2 · s2/ft
Ke = 4.42 · 104 lb·ft Ans
1
SWTJC STEM – ENGR 1201
Content Goal 15
Torricelli Example
h
d3
22
2 32 2 2
2 2
Relations : FromTorricelli's Principle,
Q A 2gh A 2gh
where
m mQ( ), A(m ), g( ), h(m)
s sSolution:
(a) Verify units of Q from the formula
m m m mQ A 2gh m m m m , same as Q.
s ss s(b) Calculating the ar
2 24 2
4 22
33
ea A of a circular opening,
d 2 cm 0.02 m (Must work in units given above!)
d 0.02A 3.14 10 m
4 4
Q A 2gh
mQ 3.14 10 m 2 9.81 1.2m
s
mQ 1.52 10 Ans
s