Chapter Two: Introduction to Engineering Calculations 2.1 Units & Dimensions Dimension : a property that can measured, length, time, mass, or temperature, or calculated by multiplying or dividing other dimensions, such as length/time (velocity), Length3 (volume), or mass/ Length3 (density).

Unit : Specific value of Dimension that have been defined by convention, custom, or law, such as gm for mass, seconds for time, & cm or ft for length.

Units treated like algebraic variables when quantities are added, subtracted, multiplied, or divided. 3 cm – 1 cm = 2 cm (3x – x) = 2x) 3 cm – 1 mm (or 1 s) = ? (3x – y = ?) Numerical values & their corresponding units may always be combined by multiplication or division. 3 N x 4 m = 12 N.m 5.0 km / 2.0 h = 2.5 km / h 7.0 km / h x 4 h = 28 km 3 m x 4 m = 12 m2 6 cm x 5 (cm / s)= 30 cm2/s

2.2 Conversion of Units A measured quantity expressed in terms of any units having appropriate dimension. Equivalent between two expressions of same quantity defined in terms of a ratio: Ratios of form of above equations are known as conversion factors.

Example: Convert an acceleration of 1 cm/s2 to its equivalent in km/yr2

Note: raising a quantity to a power raises its units to same power

2.3 Systems of Units A system of units has the following components: Basic Units: mass, length, time, temperature, electrical current, & light intensity. Multiple Units: defined as multiples or fractions of base units as minutes, hours, & milliseconds.

Derived Units: obtained in one of two ways: a) By multiplying & dividing base or multiple units (cm2, ft/min, kg.m/s2, etc.). Derived units of this type are referred to as compound units. b) As defined equivalents of compound units (e.g., 1 erg= (1 g.cm/s2), 1 Ibf ≡ 32.174 Ibm.ft/s2).

"Système Internationale d΄Unités, or SI for short, gained widespread acceptance in scientific & engineering community. Meter (m) for length, kilogram (kg) for mass, & second (s) for time. Prefixes used in SI to indicate powers of ten.

Table SI & CGS Units Base Units Quantity Unit Symbol Length meter (SI) Centimeter (CGS) m cm Mass kilogram (SI) gram (CGS) kg g Moles gram-mole mol or g-mole Time second s Temperature kelvin K Electric current ampere A Light intensity candela cd

Multiple Unit Preferences tera (T)= 1012 giga (G)= 109 mega (M)= 106 kilo (k)= 103 centi (c)= 10-2 mili (m)= 10-3 micro (μ)= 10-6 nano (n)= 10-9

Base units of American engineering system: foot (ft) for length, pound-mass (Ibm) for mass, & second (s) for time. Factors for converting from one system of units to another determined by taking ratios of quantities.

Derived Units Quantity Unit Symbol Equivalent in term of base units Volume Liter L 0.001 m3 Force (SI) Dyne (CGS) N 1 kg.m/s2 1 g.cm/s2 Pressure Pascal (SI) Pa 1 N/m2 Energy, work joule (SI) erg (CGS) gram-calorie J 1 N.m = 1 kg.m2/s2 1 dyne.cm = 4.184 kg.m2/s2 4.184 J= 1 kg.m2/s2 Power watt W 1 J/s = 1 kg.m2/s3

Multiple Unit Preferences tera (T)= 1012 giga (G)= 109 mega (M)= 106 kilo (k)= 103centi (c)= 10-2 mili (m)= 10-3 micro (μ)= 10-6 nano (n)= 10-9

Example: Convert 23 Ibm.ft/min2 to its equivalent in kg.cm/s2.

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2.4 Force & Weight According to Newton’s second law of motion, force is proportional to product of mass & acceleration. Units are, kg.m/s2 (SI), g.cm/s2 (CGS), & Ibm.ft/s2 (American engineering).

In American engineering system, derived force unit: pound-force (Ibf) defined as product of a unit mass (1 Ibm) & acceleration of gravity at sea level & 450 latitude, which is 32.174 ft/s2: 1 Ibf ≡ 32.174 Ibm.ft/s2

Example, force in Ibf required accelerating a mass of 4 Example, force in Ibf required accelerating a mass of 4.00 Ibm at a rate of 9.00 ft/s2 is

Symbol gc is sometimes used to denote conversion factor from natural to derived force unit

weight of an object is force exerted on object by gravitational attraction. Weight, mass, & free-fall acceleration of object are related by following equation:

Example: Water has a density of 62. 4 Ibm/ft3. How much does 2 Example: Water has a density of 62.4 Ibm/ft3. How much does 2.000 ft3 of water weight (1) at sea level & 450 latitude & (2) in Denvor, Colorado, where the altitude is 5374 & the gravitational acceleration is 32.139 ft/s2?

weight of the water is: 1-At sea level g=3.174 ft/s2, W=124.8 Ibf Solution: mass of the water is: weight of the water is: 1-At sea level g=3.174 ft/s2, W=124.8 Ibf 2-In Denvor, g=32.139 ft/s2, W=124.7 Ibf

2-6 Dimensional Homogeneity & Dimensionless Quantities Units & dimensions quantities can be added & subtracted only if their units are same. If units are same, it follows that dimensions of each term must be same.

For example, if two quantities expressed in terms of g/s, both must dimension (mass/time). Every valid equation must be dimensionally homogenous: that is, all additive terms on both sides of equation must have same dimensions.

Consider the equation Equation is dimensionally homogenous, since each of term u, u0, & gt has same dimensions (length/time).

For example, suppose that in dimensionally homogenous equation u = u0 + gt it is desired to express time (t) in minutes & other quantities in units given above. Equation can be written as:

1- If equation is valid, what are dimensions of constants 3 & 4? Example: Consider this equation D(ft) = 3t(s) + 4 1- If equation is valid, what are dimensions of constants 3 & 4? 2- If equation is consistent in its units, what are units of 3 & 4? 3-Derive an equation for distance in meters in terms of time in minutes

Solution: 1- Constant 3 must have dimension Length/time, & 4 must have dimension Length. 2- For consistency, constants must be 3 ft/s & 4ft. 3- Define new variables D'(m) & t'(min). The equivalence relations between old & new variables are:

Define new variables D'(m) & t'(min) Define new variables D'(m) & t'(min). The equivalence relations between old & new variables are:

Substitute these expressions in original equation: & simplify by dividing through by 3.28

Example: A quantity k depends on temperature T in following manner: Units of quantity 20,000 are cal/mol, & T is in K (Kelvin). What are units of 1.2x105 & 1.987?

Solution: Since equation must be consistent in its units & exp is dimensionless, 1.2x105 should have same unit as k, mol/(cm3.s). Moreover, since argument of exp must be dimensionless, we can write:

Answers are thus 1.2 x 105 mol/(cm3.s) & 1.987 cal/(mol.K) 2-2-2009 (All units cancel) Answers are thus 1.2 x 105 mol/(cm3.s) & 1.987 cal/(mol.K) 2-2-2009