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# 1 dimensions and units

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### 1 dimensions and units

1. 1. 1 DIMENSIONS AND UNITS Definition: Dimensions are basic concepts of physical measurements such as: – Length = [L] – Time = [T] – Mass = [M] – Temperature = [θ] Units are terms that precede and describe the dimensions.
2. 2. 2 Classification of dimensions Definition: Fundamental or basic dimension – dimensions that are measured independently and enough to express essential physical quantities Derived dimensions – dimensions that are products or quotients of fundamental dimensions
3. 3. 3 Systems of units SI (Le Systeme Internationale d’Unites) system • Simple system because fewer names are associated with the dimensions. • The current metric system. • Use prefixes (e.g., c, M, n, m) and are factors of 10. AE (American Engineering) system • Deeply rooted in the United States. • Other names of this system are English, U.S. Customary or Imperial System.
4. 4. 4 Countries that do not use SI: Liberia, Myanmar and United States
5. 5. 5 SI dimensions and units Source: Himmelblau, D.M. & Riggs, J.B., 2004
6. 6. 6 AE dimensions and units Source: Himmelblau, D.M. & Riggs, J.B., 2004
7. 7. 7 Some important tips about units: – Uppercase and lowercase letters should be strictly followed, e.g. K (kelvin), Pa (pascal), L (liter). – Unit abbreviations have the same form for both singular and plural and NOT followed by a period (.) except for inches (in.). – Multiplication of two or more units will combine those two or more units together separated by a period (.) e.g. m.s. – Hyphen (-) should NOT be used in combination of units. – Dot (.) in multiplication of numbers should be AVOIDED such as 2 . 5. – Commas in numbers (e.g. 100,000) should also be AVOIDED.
8. 8. 8 Mathematical operations with units • Addition, subtraction, equality • Add, subtract, or equate numerical quantities only if they are of the same units. • E.g., 5 kg + 10 J are not of the same units, thus cannot be added. • E.g., 10 lb - 10 g can be subtracted only after the units have been converted to be same units. • Multiplication and division • Multiplication and division can be done on unlike units but cannot be cancelled or merged if they are different. • E.g., 200 (kg)(m)/(s2) cannot be cancelled or merged because the units are different from each other
9. 9. 9 Handling mathematical operations: sin, cos, log and e – The variable that the mathematical operation is applied on must be converted to dimensionless form first. Example 1 D = 24.5 – 24.3e-0.31t t < 150 s » where D is in meters (m) and t is in time (s). What is the units of the constants 24.5 and 0.31 respectively? » The unit of 24.5 is meter (m) and the unit of 0.31 must be s-1.
10. 10. 10 Conversion of units and conversion factors • As a future scientist, technologist, or engineer, you must pay close attention to your units. • The procedure for converting a set of units to another is by multiplying the number and its units to the ratio required (a.k.a. conversion factor) • Grid method is a simple method to use to avoid confusion when converting units. • Examples of conversion factors: » 1 m = 100 cm 1 m / 100 cm or 100 cm / 1 m » 4.45 N = 1lbf 4.45 N / 1 lbf or 1 lbf / 4.45 N
11. 11. 11 Example 2 Convert from 328 ft/s to mi/h. You need to know the required conversion factors such as, • 1 mi = 5280 ft • 1 min = 60 s • 1 h = 60 min Using the grid method, 1 h1 min5280 fts 60 min60 s1 mi328 ft = 234 mi/h
12. 12. 12 Example 3 Convert from 452 cm/s2 to m/min2. You need to know the required conversion factors such as: • 1 m = 100 cm • 1 min = 60 s (1 min)2100 cms2 (60 s)21 m452 cm = 16272 m/min2
13. 13. 13 Pound mass (lbm) and pound force (lbf) Newton’s 2nd law (SI system) for weight F = Cma Where, F = force C = constant m = mass a = acceleration • In the SI system, force of 1 N is where 1 kg is accelerated at 9.8 m/s2; C has to be 9.8 (N)/[(kg)(m)/s2] s2 s2(kg)(m) 9.8 m1 kg1N F = = 9.8 N
14. 14. 14 Newton’s 2nd law (AE system) for weight •lbf and lbm can be the same value if it is at Earth’s surface •Mass of 1 lbm is accelerated at g ft/s2 (= 32.2 ft/s2) • is a constant •lbf and lbm are not the same units •1 lbf ≈ 4.44822 N s232.174(lbm)(ft) g ft1 lbm1(lbf)(s2) F = = 1 lbf 32.174 lbm ft lb f s 2
15. 15. 15 Dimensional consistency •A basic principle states that equations must be dimensionally consistent. •Using van der Waal’s equation as an example, Example 4 What are the dimensions of a and b? – ‘a’ has the units (pressure)(volume)2 – ‘b’ has the same units as ‘V’ [volume] Dimensionless numbers •There are some variables or group of variables that do not have a net unit. These are called non-dimensional or dimensionless variables, for example, gcm3s (cm)(s)gcmcm  Dv NRE    RTbV V a p        2
16. 16. 16 Significant figures Any meaningful value have 3 types of information associated with it: 1. the magnitude of the variable being measured. 2. its units. 3. an estimate of its uncertainty.
17. 17. 17 Example 5 • The number 140.06 have 5 significant figures • 140.06 lies in the uncertainty interval of • 140.06 ± 0.005 • From 140.055 to 140.065 • If a number is displayed as 130.000, it means that the number is more accurate since it contains 6 significant figures. Multiplying or dividing numbers • A very important tip is to keep the final answer the lowest number of significant figures when multiplying or dividing. Example 6 40.392 × 87.0345 ÷ 0.32 = 11000 (2 s.f.)
18. 18. 18 Adding and subtracting numbers • When adding or subtracting, the significant figure that should be kept in the final answer must be determined by the largest error interval. For example, Example 7 125.8 + 0.045 = ? Error intervals of 125.8 and 0.045 are: • 125.8 ± 0.05 and 0.045 ± 0.0005 • The larger error of 125.8 obscures the error of 0.045 • Thus,125.8 + 0.045 = 125.845 = 125.8 (4 s.f.) • This is because the final summation should account for only the larger error of 0.1 from 125.8
19. 19. 19 Something to think about, • Avoid increasing the precision (number of significant figures) of the final answer when compared to the values used in the calculations. • One or two figures can be used in the intermediate calculations. • Numbers such as 1 kg or 20 cm can be assumed that its number of significant figures are high (such as 1.000 kg or 20.000 cm). They are called PURE or DEFINED numbers, such as 3 cars or 2 apples, and sometimes dimensionless.
20. 20. 20 Example 8 Calculate the following, giving the accurate number of s.f. in each final answer. Tip: Keep the same number of decimal places as the number with the least amount of decimal places. • 1.421 + 0.4372 = • 0.0241 + 0.11 = • 0.14 + 1.2243 = • 760.0 + 0.011 = • 1.0123 – 0.002 = • 123.69 – 20.1 = • 463.231 – 14.0 = • 47.2 – 0.01 =