US, Burma, Liberia are three countries that have not adopted the SI as the sole system of units (from Wikipedia)

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1 1.2 System of Units A single system is efficient, convenient, economical, and makes sense. The US system units is difficult for the United States to give up because a lot of expert technical knowledge and intuition gained over generations is tied to this particular unit system. The world will use a single system once the US decides it wants too give up the US system. The intent has been there for decades but the US units are still used vigorously. Living 2 miles from the grocery store is better imagined than 3.21 kilometers for persons living in the US. There are two popular Systems of Units in practice. The first one is the System International or SI System of Units used across the world where the unit for weight is Newtons. The second is the US or the English System mostly used in USA where the unit for weight is pound (or pound force to be strict). Any material for sale in the USA will display the specifications in both system of units. Figure Widespread use of US Units US, Burma, Liberia are three countries that have not adopted the SI as the sole system of units (from Wikipedia) SI - or System International The International system of units is used globally for commercial purposes as well as in science and engineering. It appeared in 1960 and some of the standards are maintained at the International Bureau of Weights and Measures at Severn, near Paris, France. It is also sometimes referred to as the metric system. The SI is a more complete system. In fact the US/English system have no units for many of the physical quantities that have been defined or discovered recently. The SI is a system of units of measurement defined through seven (7) BASE UNITS Table SI Base Units Name Unit Symbol Quantity Symbol metre/meter m length l kilogram kg mass m second s time t ampere A electric current I kelvin K thermodynamic temperature T candela cd luminous intensity I v

2 mole mol amount of substance n SI: Base and Derived Units The meter stick is an instrument that measures one meter between marked ends. Figure The meter stick In SI units the measured length is 1 meter. The information on length can be expressed in several alternate units in the same unit system. In the same SI units it is also : 100 centimeters 1000 millimeters 10 6 microns 10 9 nanometers It is important to use the appropriate unit defined through context and practice. For example the nanometer is useful in discussing the wavelength of light. Microns may be appropriate when discussing products in the semiconductor industry. Millimeters may be appropriate for discussing blood vessels. The BASIC unit of length is the meter (alternately metre) In addition to BASIC units many physical quantity can be expressed through DERIVED units. Derived units imply a combination of basic units. For example The density of water is 1000 kilogram/meter 3 [kg/m 3 ]. This is a derived unit. Every physical quantity can be associated with a corresponding basic or derived unit in the SI system of units. Consider another example of the linear elastic spring from physics. We will apply a force (F) to stretch it through a distance (s). Not only is the force equal to spring constant (k) times the deflection (s), the units for the force must equal to the product of the units for the spring constant and the deflection in base units. F = k s or [N] = [k] [s] k = F/s or [k] = [N/m]

3 Figure A linear spring - derived units Units used in Mechanics Table SI Units in Mechanics Quantity Unit Unit Symbol Alternate Symbol length meter m mass kilogram kg time second s acceleration meter per second m/s 2 squared angle radian rad area square meter m 2 density kilogram per cubic meter kg/m 3 energy Joule J 1 Joule = 1Nm force Newton N 1 N: 1 kg-m/s 2 moment of a force (torque) Newton meter Nm pressure Newton per square metern/m 2 Pa: Pascal stress Newton per square metern/m 2 Pa: Pascal volume (solid) cubic meter m 3 volume (liquid) liter 10-3 m 3 L work Joule J 1 Joule = 1Nm There are two corollaries that we will respect: 1. Any law or equation we develop must have the same unit on both side of the equal sign 2. Every term in the equation must also have the same units US System of Units The US customary unit system has some overlap with the British Imperial system. It was defined in modern form in In 1988 Congress passed a bill for confirming adoption of the SI system but left the timing indefinite. So the adoption is in transition. We will refer to this system as the US system. We can recognize base units and derived units like we did for SI units. We also need a way to translate information between the two systems. However the US system does not have units available

4 for a lot of physical quantities that have been important recently. Unlike the SI system where units for the same quantity, for example length, can vary by orders of 10 (millimeter, centimeter, meter, kilometer), the units for the same quantity in the US have no connection. The units for length in the US system of units can be expressed in several ways. This is displayed in Table Table Various US units for length (source Wikipedia) Unit Alternate Unit SI Equivalent 1 inch (in) 2.54 cm 1 foot (ft) 12 in m 1 yard (yd) 3 ft m 1 mile (mi) 1760 yd km Survey 1 link (li) ft or 7.92 in m 1 (survey) foot (ft) m m 1 rod (rd) 25 li or 16.5 ft m 1 chain (ch) 4 rd m 1 furlong (fur) 10 ch m 1 survey (or statute) mile 8 fur km (mi) 1 league (lea) 3 mi km Nautical 1 fathom (ftm) 2 yd m 1 cable (cb) 120 ftm or fur m 1 nautical mile (NM or nmi) cb or mi km Similar tables can be established for area, volume, mass, cooking measures etc. US: Base and Derived Units Table lists some of the base and derived units in the US system. It also shows the corresponding SI Units and the corresponding conversion factors for equivalence. Table US Units in Statics and the corresponding SI Units with conversion factors Quantity US Unit Equivalent SI Unit length 1 feet [ft] [m] mass 1 slug [slug] [kg] mass 1 pound mass [lbm] [kg] time 1 second [s] 1 [s] acceleration 1 [ft/s 2 ] [m/s 2 ] angle 1 [rad] 1 [rad] area 1 [ft 2 ] [m 2 ] density 1 [slug/ft 3 ] [kg/m 3 ] energy 1 [ft-lb] [J ] force 1 pound force [lb] [N] moment of a force (torque) 1 [lb-ft] [Nm] pressure 1 [lb/ft 2 ] [Pa] pressure 1 [lb/in 2 ], [psi] x10 3 [Pa] Stress 1 [lb/in 2 ], [psi] x10 3 [Pa] volume (solid) 1 [ft 3 ] [m 3 ] volume (liquid) 1 gallon [gal] Liter [L] work 1 [ft-lb] [J or Nm]

5 Conversion between Systems of Units To work in both system of units you need to how to convert between them. It is a simple arithmetic process with both the numbers and the units themselves. For example, in USA, the speed at which you are traveling is usually expressed in miles per hour say 25 [mph]. In basic units you will need to express the speed in feet per second [ft/s]. In the SI system the corresponding base units are meter per second [m/s], while the popular expression for the speed is kilometer per hour [km/hr]. In order to convert from [mph] to feet per second [ft/s] it we need to know that 1 [mile] is 5280 feet [ft] 1 [hour] is 3600 second [s] To convert from US units [mph] to SI units [m/s] To from [m/s] to [km/hr] or [kph] we can use the conversion factor listed in Table Accuracy For the conversion of units and other calculations in this book you are likely to use a calculator and sometimes you will get a long number. Further additional operations with the number you will continue to obtain these long numbers. For example: cos (25) = and 2*cos(25) = Do we need to know the answer to so many places after the decimals? Let us say you are expressing your lengths in centimeters. If you are building desk and chairs you need one place after the decimal for a well built product. If you are building cars you need two places after the decimal for a good looking car. If you are building airplanes you need three places after the decimal for a well functioning aircraft. The number of decimal points is related to the accuracy of your calculations. In engineering problems the information is usually considered to be accurate to 0.2%. For example if you are using 2500 [N] for the weight of the pumpkin, a 0.2% error in this estimate is expressed as 2500 ± 5 [N]. Your calculations are only as good as the accuracy of your information. It does not make sense to report answers to the ninth place after the decimal. In this ebook we will report our calculations to two places after the decimal unless otherwise warranted. You can choose also to round off your value to the second decimal place. If the number in the third place after the decimal is greater than or equal to 5 then the second place number is increased by 1,

6 except if the second place number is 9. You will then have to change the first place number or not round off. You must be consistent in applying round off. Hence cos (25) = 0.91 and 2*cos(25) = Dimension How do we know that the units for acceleration in basic SI units is [m/s2]? We can always look it up in a table (Table 1.2), in a book,or obtain it by searching the Internet. A more fundamental idea is that every physical quantity is associated with a Dimension. This is very important in the subject of fluid mechanics. Also, this is very different from the concept of Dimension (D) that is geometrically associated with a problem (like 1 D problem, 2D problem, or 3D problem) in science and engineering, that progressively includes more information, more equations, more work, and is more difficult to solve. We will come this path in this book later. A simple way to understand Dimension that is associated with the units of a physical quantity is to accept that are a set of basic dimensions (Table 1.2.5). Table Basic Dimensions Quantity Symbol length L mass m time t electric current I thermodynamic temperature T luminous intensity I v amount of substance n Let us consider speed as an example. We know that the average speed is computed by dividing the distance traveled by the time for the travel. The dimension for speed is [L/t]. The units for speed can be obtained by substituting appropriate units for the dimensions in the definition. Units for speed : [m/s]; [km/h]; [ft/s]; [ft/h]; [miles/h]; [in/s]; etc. Similarly acceleration is Units for acceleration: [m/s 2 ]; [ft/s 2 ]; [in/s 2 ]; etc. Example Using the Newton s law of gravitation, find (a) the dimension of the universal gravitational constant (G); (b) Its basic units in the SI system; (c) Its basic unit in the US system. The Newton s law of gravitation, where F is the force between the two masses, m1 and m2, and r is the distance between the centers, can be expressed as:

7 Assumptions: To proceed we need to know the dimension of the force. We can look up a book or we can obtain it by using Newton s second law, F = ma (from physics). Solution: Dimension of force, F: (a) Dimension of G : (b) [G] = [m 3 /kg-s 2 ] or [m 3 kg -1 s -2 ] (c) [G] = [f t3 /slug-s 2] = [ft 3 slug -1 s -2 ] Note: We work with symbols the same was as we work with numbers.