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Review SI Units • • • • • Time: Mass: Distance: Temperature: Charge: Important Secondary Quantities • Force: • Energy (work): • Power: Questions: • What is a kilowatt-hour? • Where is it used? In-Class Activity #1 A study of households and businesses in the Boston, Massachusetts area found that air conditioning units were used for an average of 4600 hours per year. Determine the total annual cost of electricity required to operate a 10,000 Btu/hr air conditioning unit in the Boston area if the electric rate is $ 0.071/kWhr. Notation • The units of variables will be referred to by putting the variables in brackets • Consider the equation: v-vo = a·t • What is the equation saying? • [v] will refer to the dimension of v [t] will refer to the dimension of t • In order to determine the units of an unknown (say a), we need to be able to write equations in terms of units, e.g. v vo at Rules of Homogeneity • Definition: An equation is said to be dimensionally homogeneous if all terms separated by plus and minus and equal sign have the same dimension. • Consider the previous example: v vo at • In order to be homogeneous, v vo a t Rule 1 • If the dimensional quantities are replaced by their primary units the equation should reduce to an identity. • In our example, what are the primary units? v vo a t Rule 2 • Dimensions do NOT add or subtract. In order to add or subtract variables, they must have the SAME units. • In our example, [v] = [vo]. If you subtract them you have no units on the left Rule 3 • Dimensions DO multiply and divide. In-Class Activities #2, #3 • If P1 = 400 W and P2 = 12 Btu/minute, what is P1+P2 in SI units? • Which, if any, of the following are correct for acceleration units: m m a) m/s/s b) s c) ss s m 1 d) s s Exercises • A relationship between Force F in (N), distance x in (m), mass M in (kg) and speed v in (m/s) is suggested as 1 2 Fx Mv 2 Is this equation dimensionally homogeneous? Exercises (cont) • A relationship between Force F in (N), time t in (s), mass M in (kg) and speed v in (m/s) is suggested as: Ft2 = Mv Is this equation dimensionally homogeneous? Prefixes • As you’ve already seen, we’ll deal with both very large numbers and very small numbers. • We will use scientific notation and/or engineering prefixes to represent these numbers Refresher: common prefixes T G M k tera giga mega kilo 1012 109 106 103 d c m μ n p deci centi milli micro nano pico 10-1 10-2 10-3 10-6 10-9 10-12 Examples in Engineering Notation • 2x103 Volts = • .00045 A = • 1.3x10-6 C = • 10x107 Hz = Some Quantities Must Be Dimensionless • • • • In Calculus you will see eax ax must be dimensionless (no units) 1/1 Example, if x is in m, a is in 1/m If x is in s, a is in 1/s • Some dimensionless quantities need units to make sense • Example if v = vo e-at v and vo have units of volts Class Activity In an electrical circuit the current i(t) in A (Amperes) changes with time t in s according to the function i(t) = e-2t. a) How could this function be dimensionally consistent knowing that the exponential function is always dimensionless (RHS), and that i(t) is in A on the LHS? b) What are the units of the constant 2 in the exponential function? More on Units • Remember x = cos(θ)? • What are the units of θ? Angular frequency • What are the units of ω in y(t) = cos(ωt) if t has units s? • ω is called the angular frequency • ω = 2pf