1 Introduction to Engineering Spring 2007 Lecture 12: Units & Measurements

2 Review Cash Flow Net Present Value Profits

3 Review - Net Present Value The Net Present Value, NPV, is the value now of some amount that will be realized at some future time. where A t is the cash flow at time t k is the expected rate of return

4 Outline Introduction to Dimensions & Units Other Systems Dimensions in Equations

5 Introduction to Dimensions and Units

6 Goal  Describe the basic techniques for the handling of units and dimensions in calculations. Describe the basic techniques for expressing the values of process variables and for setting up and solving equations that relate these variables.

7 Definitions Dimensions are properties that can be measured such as length, time, mass, temperature, or calculated by multiplying or dividing other dimensions, such as velocity (length/time) Units are means of expressing the dimensions such as feet or meter for length, hours/seconds for time. Every valid equation must be dimensionally homogeneous: that is, all additive terms on both sides of the equation must have the same unit

8 Important???

9 What does length mean? How long is this rod? Start with a rod of known length – 1 m How do we know that the measuring rod was 1 m? What is a meter anyway?

10 Standard for length “the length of the path traveled by light vacuum during a time interval of 1/ seconds” 1 meter t = 1/ s t = 0 s Laser

11 Other Fundamental Units Time(sec) “the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-113 atom” Mass (kilogram) a cylinder of platinum-iridium alloy maintained under vacuum conditions by the International Bureau of Weights and Measures in Paris Temperature (Kelvin) The kelvin unit is 1/ of the temperature interval from absolute zero to the triple point of water. The triple point of any substance is that temperature and pressure at which the material can coexist in all three phases (solid, liquid and gas) in equilibrium

12 International System of Units 1 PrefixDecimal MultiplierSymbol Atto Femto pico nano micro milli centi deci afpnmcdafpnmcd

13 International System of Units 2 PrefixDecimal MultiplierSymbol deka hecto kilo mega Giga Tera Peta exa da h k M G T P E

14 Rules for Dimension Symbols 1 Periods are never used after symbols Unless at the end of the sentence These symbols are not abbreviations Use lowercase letters unless the symbol derives from a proper name m, kg, s, mol K, Hz, Pa (Pascal), C (Celsius)

15 Rules for Dimension Symbols 2 Symbols rather than self-styles abbreviations always should be used A (not amp), s (not sec) An s is never added to the symbol to denote plural A space is always left between the numerical value and the unit symbol 43.7 km (not 43.7km) 0.25 Pa (not 0.25Pa) Exception; 5 0 C, 5’ 6”

16 Derived Units Some fundamental units are defined in terms of other units For example, force is defined in terms of Newtons based on the physical relationship between force, mass and acceleration: F = ma So 1 Newton is the amount of force required to accelerate a 1 kg mass to 1 meter per second squared 1 N = 1 kg 1 m 1 s 2

17 Other Systems

18 US Customary System (USC) Based on things that made sense to people Previously known as English (or British) 1 inch = 3 dry, round, barleycorns end-to-end 1 foot = length of King Edward I’s foot 1 mile = 1000 double paces of Roman soldier 12 in/ft; 4 in/hand; 3 ft/yd; 5280 ft/mile

19 Systeme Internationale (SI) Sometimes called the metric system, although different Attempted to be less arbitrary Example: 1 meter original: one ten-millionth of the distance from the equator to either pole current: based on wavelength of light Based on powers of 10

20 American Engineering System of Units (AES) Fundamenal DimensionBase Unit length [L] mass [m] force [F] time [T] electric change [Q] absolute temperature [  luminous intensity [l] amount of substance [n] foot (ft) pound (lb m ) pound (lb f ) second (sec) coulomb (C) degree Rankine ( o R) candela (cd) mole (mol)

21 Conversion between Systems For length the relationship is: To convert from feet to meters use: To convert from meters to feet use: 5 ft = 5 ft x = m 1 ft m Conversion Factor 5 m = 5 m x = ft m 1 ft

22 Using MatLab A conversion program is easy to write in MatLab:

23 Example Run

24 Dimensions in Equations

25 Dimensional Analysis Physical laws must be independent of arbitrarily chosen units of measure. Nature does not care if we measure lengths in centimeters or inches or light-years or … Check your units! All natural/physical relations must be dimensionally correct. Dimensional Analysis refers to the physical nature of the quantity and the type of unit (Dimension) used to specify it. Distance has dimension L. Area has dimension L 2. Volume has dimension L 3. Time has dimension T. Speed has dimension L/T

26 DA of Gulliver’s Travels Gulliver was 12x the Lilliputians How much should they feed him? 12x their food ration? A persons food needs are related to their mass (volume) – This depends on the cube of the linear dimension.

27 Calculations 1 Define the relative size Let L G and V G denote Gulliver’s linear and volume dimensions. Let L L and V L denote the Lilliputian’s linear and volume dimensions. Since Gulliver is 12x taller than the Lilliputians, L G =12 L L Now V G  (L G ) 3 and V L  (L L ) 3

28 Calculations 2 VGVG VLVL = LL3LL3 LG3LG3 = LL3LL3 (12L L ) = = RESULT: Gulliver needs to be fed 1728 times the amount of food each day as the Lilliputians. This problem has direct relevance to drug dosages in humans

29 Dimensions in Equations 1 Rule 1 - All terms that are added or subtracted must have same dimensions Rule 2 - Dimensions obey rules of multiplication and division All have identical dimensions

30 Dimensions in Equations 2 Rule 3 - In scientific equations, the arguments of “transcendental functions” must be dimensionless. X must be dimensionless in the following expressions: A = sin(x) B = log(x) C = exp(x) D = 3 x

31 Possible Quiz Remember that even though each quiz is worth only 5 to 10 points, the points do add up to a significant contribution to your overall grade If there is a quiz it might cover these issues: Why are Dimensions and Units important? What is the relationship between physical laws and units? What was USC based on?