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 Rule 2.Algebraic Manipulation Rule 3.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 = v 0. t + (1/2). a. t where d (m), v 0 (m/s), t (s), and a (m/s 2 ) m = (m/s). s + (none). (m/s 2 ). s m = m + m/s A problem? The relation is inconsistent! Formula is incorrect! Distance formula:d = v 0. t + (1/2). a. t 2 m = (m/s). s + (none). (m/s 2 ). s 2 m = m + m The formula is consistent! Examples dimensionless constant (no units!)

SWTJC STEM – ENGR 1201 DimAnalysis cg13d Particle Energy Example

SWTJC STEM – ENGR 1201 DimAnalysis cg13d Particle Energy Example

SWTJC STEM – ENGR 1201 DimAnalysis cg13d Particle Energy Example

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) e x 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) = No units!

SWTJC STEM – ENGR 1201 Content Goal 15 Richter Scale Example Earthquake intensity is measured on the Richter Scale. M R = log(A) where is a seismic amplitude factor. The famous San Francisco earthquake of 1906 was M R = 7.8 on the Richter scale. Does A have units? No! According to Rule 3, Transcendental Function Does M R have units? No! Ditto.

SWTJC STEM – ENGR 1201 Content Goal 15 Bernoulli Example

SWTJC STEM – ENGR 1201 Content Goal 15 Bernoulli Example

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 K e = W · v 2 / (2 · g) where K e (lb·ft), W (lb), v (ft/s), g (32.2 ft/s 2 ) What is the system of units? Related to USCS: 1 ton = 2000 lbs, 1 knot = ft/s Is USCS coherent?Yes. What is the relation for kinetic energy? Related to USCS: 1 ton = 2000 lbs, 1 knot = ft/s

What is first step? SWTJC STEM – ENGR 1201 Content Goal 15 Coherent System Example K e = W · v 2 / (2 · g) where K e (lb·ft), W (lb), v (ft/s), g (32.2 ft/s 2 ) Convert to base units. W in tons and v in knots. 20 tons · 2000 lbs/ton = 40,000 lbs = 4 · 10 4 lbs What’s not in base units? 5 knots · (ft/s)/knot = 8.44 ft/s Substituting, K e = 4 · 10 4 lb · (8.44 ft/s) 2 / (2 · 32.2 ft/s 2 )

SWTJC STEM – ENGR 1201 Content Goal 15 Coherent System Example K e = 4 · 10 4 · / 64.4 lb · ft 2 /s 2 · s 2 /ft K e = 4.42 · 10 4 lb·ft  Ans 1

SWTJC STEM – ENGR 1201 Content Goal 15 Torricelli Example h d