ENGINEERING MATHSCompiled and presented by Doren Nedrick

Squares of NumbersWhen a number is multiplied by itself, the product is called the square of the given number. Recall 22, read 2 squared means 2 x 2 = 4, so the square of 2 is 432 = 9 so the square of 3 is 942 = 16 so the square of 4 is 1652 = 25 so the square of 5 is 25.

Finding the Square Root of a NumberIf a number is a product of 2 identical factors either factor is called a square root of the number.Example: 49 = 7 x 7 thus 7 is the square root of 49 is the symbol for square root.49 = 7; 25 = 5; b2 = b

EquationsConsider the statement: 3x + 6 = 18This is an equation. It means that the left hand side is equal to (i.e. the same value as) the right hand side.

Solving EquationsSolving an equation means finding the number that the letter stands for so that the two sides are equal.

Substituting numbers into a FormulaGiven that Rf = R0 (1 + t)Find Rf when R0 = 100 = 0.004/oC and t = 35oCRf = R0 (1 + t) = 100 (1 + 0.004 X 35) = 100 (1 + 0.14) = 100 (1.14) = 100 x 1.14 = 114

Changing the Subject of a formulaConsider the formula: RT = R1 + R2Make R1 the subject of the formulaTake R2 from each side gives RT R2 = R1 + R2 R2 RT R2 = R1i.e. R1 = RT R2

Example 1The total resistance of two resistors connected in series is 75. The value of one resistor is 33. What is the value of the other one?

Example 2Consider POUT = PIN PLOSS (make PIN the subject of the formula)Add PLOSS to both sides give POUT + PLOSS = PIN PLOSS + PLOSSPOUT + PLOSS = PIN i.e. PIN = POUT + PLOSS

ActivityA motor develops 3 000W of power. Calculate the input power to the motor if it loses 350W due to mechanical loss, copper and hysterisis loss. POUT = PIN PLOSS3000W = PIN 350W (add 350W to 2 sides)3000W + 350W = PIN - 350W + 350W3 350W = PIN Hence PIN = 3 350W

Example 3V = I x R same as V = IRDividing both sides by R V = I x RR RWell achieve V = I R i.e I = V R

ExerciseCalculate the current flow through a circuit containing a resistance of 3 and supplied by a 12V battery.R = 3V = 12VI = ?V = I x R12V = I x 3

SolutionV = I x R12V = I x 3 Divide both sides by 312V = I x 33 34A = I Hence I = 4A

Example 4Consider P = E t Solve for EMultiply both sides by t well achieve P x t = E x 1 t tP x t = E i.e. E = P x t

IndicesIn the number 23, the 3 is called the index or power and 23 means 2 x 2 x 2.Thus indices are a kind of shorthand notation. All other forms of indices are derived from this. 23 = 8

Multiplication of IndicesWe can multiply one number to a power by the same number to another power by adding the powersExample: 52 x 54 = 5 x 5 x 5 x 5 x 5 x 5 = 56

Division of IndicesWe can also divide one number to a power by the same number to another power by subtracting the powersExample: 25 22 = 23 25 22 = 2 x 2 x 2 x 2 x 2 2 x 2

Zero IndicesConsider a3 a3 = Subtracting indices give a3 a3 = a0 = a x a x a a x a x a Dividing gives a3 a3 = 1 a0 = 1 i.e. (any number)0 = 1

Negative IndicesNow consider a3 a5Subtracting indices gives a3 a5 = a-2a3 = . a x a x a . = 1a5 a x a x a x a x a a2 Therefore a-2 = 1 a2

Activity102 x 103 = 1 103 = 10-6 1 =10-6 10-9 =

Scientific notationIn many disciplines of science and engineering, very large and very small numerical quantities must be managed. Take for example the mass of a proton, one of the constituent particles of an atoms nucleus:Proton mass = 0.00000000000000000000000167 grams

Standard FormOr, consider the number of electrons passing by a point in a circuit every second with a steady electric current of 1 amp:1 amp = 6,250,000,000,000,000,000 electrons per secondA lot of zeros, isnt it? Obviously, it can get quite confusing to have to handle so many zero digits in numbers such as this, even with the help of calculators and computers.

Standard form0.00000000000000000000000167 = 1.67 x 10-24

6,250,000,000,000,000,000 = 6.25 x 1018

SummaryEngineering Metric PrefixesCan you name the prefixes and their meaning?

SummaryEngineering Metric PrefixesCan you name the prefixes and their meaning?

SummaryVery large and very small numbers are represented with scientific and engineering notation.Scientific and Engineering Notation47,000,000 = 4.7 x 107 (Scientific Notation) = 47. x 106 (Engineering Notation)

Summary0.000 027 = 2.7 x 10-5 (Scientific Notation) = 27 x 10-6 (Engineering Notation)0.605 = 6.05 x 10-1 (Scientific Notation) = 605 x 10-3 (Engineering Notation) Scientific and Engineering Notation

Summary When converting from a larger unit to a smaller unit, move the decimal point to the right. Remember, a smaller unit means the number must be larger.Metric Conversions0.47 MW = 470 kW

Summary When converting from a smaller unit to a larger unit, move the decimal point to the left. Remember, a larger unit means the number must be smaller.Metric Conversions10,000 pF = 0.01 mF

Summary When adding or subtracting numbers with a metric prefix, convert them to the same prefix first.Metric Arithmetic10,000 W + 22 kW = 10,000 W + 22,000 W = 32,000 WAlternatively,10 kW + 22 kW = 32 kW

Summary When adding or subtracting numbers with a metric prefix, convert them to the same prefix first.Metric Arithmetic200 mA + 1.0 mA = 200 mA + 1,000 mA = 1,200 mAAlternatively,0.200 mA + 1.0 mA = 1.2 mA

ExerciseWhat is the total resistance of a network with 10k, 500 and a 0.04M resistor?What is the total current supplied to a parallel network of resistors if 0.1A of current flows through R1 10mA through R2 and 1000A through R3?

Multiplication of engineering notation10k x 1.2m = 10 x 103 x 1.2 x 10-3 = 10 x 1.2 = 12103 x 10-3 = 100 or 1Hence 10k x 1.2m = 12 x 1 = 12

Division of engineering notation 12 = 12 = 12 x 100 1m 1 x 10-3 1 x 10-3

12 1 = 12100 10-3 = 103

= 12 x 103 or 12k

Example 212 0.6 = 12 x 10-6 0.6 x 10012 0.6 = 2010-6 100 = 10-6

20 x 10-6 or 20

Example 3A copper cable 1500 m long (l), has a cross-sectional area () of 25 x 10-6 m2. The resistivity of copper () is 1.7 x 10-6 m. Calculate the resistance of the cableR = x l

R = x l R = 1.7 x 10-6 x 1500 25 x 10-6R = 2550 x 10-6 25 x 10-6R = 102

Exercise1. calculate the voltage drop across a resistor of 22k if it carries 12mA of current2. An insulator has 15A of current flowing through it with 4.5kV across it. What is its resistance?3. What is the resistance of an AM/FM radio if it draws 120mA from a 13.6Vdc supply.