Chapter 2. Technical Mathematics A PowerPoint Presentation by Paul E. Tippens, P
Whiteman, Bobbie, Foreign News Editor has reference to this Academic Journal, PHwiki organized this Journal Chapter 2. Technical Mathematics A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University © 2007 MATHEMATICS is an essential tool to the scientist or engineer. This chapter is a review of most of the skills that are necessary as long as underst in addition to ing in addition to applying physics. A thorough review is essential. Preparatory Mathematics Basic geometry, algebra, as long as mula rearrangement, graphing, trigonometry, scientific notation, in addition to such are normally assumed as long as beginning physics. If unsure, please at least run through the very focused review in this module.
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Objectives: After completing this module, you should be able to: Add, subtract, multiply, in addition to divide signed measurements. Solve in addition to evaluate simple as long as mulas as long as all parameters in an equation. Work problems in Scientific Notation. Construct in addition to evaluate graphs. Apply rules of geometry in addition to trigonometry. Addition of Signed Numbers To add two numbers of like sign, sum the absolute values of the numbers in addition to give the sum the common sign. To add two numbers of unlike sign, find the difference of their absolute values in addition to give the sign of the larger number. Example: Add (-6) to (-3) (-3) + (-6) = -(3 + 6) = -9 Example: Add (-6) to (+3). (+3) + (-6) = -(6 – 3) = -3 Arithmetic: Come on, man Whats up with this! I have no trouble with addition in addition to subtraction. This is grade school, man!
Example 1. A as long as ce directed to the right is positive in addition to a as long as ce to the left is negative. What is the sum of A + B + C if A is 100 lb, right; B is 50 lb, left; in addition to C is 30 lb, left. Given: A = + 100 lb; B = – 50 lb; C = -30 lb A + B + C = (100 lb) + (-50 lb) + (-30 lb) A + B + C = (100 lb) + (-50 lb) + (-30 lb) A + B + C = +20 lb Net Force = 20 lb, right A + B + C = +(100 lb – 50 lb – 30 lb) Subtraction of Signed Numbers To subtract one signed number b from another signed number a, change the sign of b in addition to add it to a, using the addition rule. Examples: Subtract (-6) from (-3): (-3) – (-6) = -3 + 6 = +3 Subtract (+6) from (-3): (-3) – (+6) = -3 – 6 = -9 Example 2. On a winter day, the temperature drops from 150C to a low of -100C. What is the change in temperature Given: t0 = + 150C; tf = – 100C Dt = tf – t0 Dt = (-100C) – (+150C) = -100C – 150C = -25 C0 Dt = -25 C0 What is the change in temperature if it rises back to +150C Dt = +25 C0
Multiplication: Signed Numbers (-12)(-6) = +72 ; (-12)(+6) = -72 If two numbers have like signs, their product is positive. If two numbers have unlike signs, their product is negative. Examples: Division Rule as long as Signed Numbers If two numbers have like signs, their quotient is positive. If two numbers have unlike signs, their quotient is negative. Examples: Extension of Rule as long as Factors Examples: The result will be positive if all factors are positive or if there is an even number of negative factors. The result will be negative if there is an odd number of negative factors.
Example 3: Consider the following as long as mula in addition to evaluate the expression as long as x when a = -1, b = -2, c = 3, d = -4. x = -1 + 48 x = +47 Working With Formulas: Many applications of physics require one to solve in addition to evaluate mathematical expressions called as long as mulas. Consider Volume V, as long as example: V = LWH Applying laws of algebra, we can solve as long as L, W, or H: Algebra Review A as long as mula expresses an equality, in addition to that equality must be maintained. If x + 1 = 5 then x must be equal to 4 in order to maintain equality. Whatever is done to one side of an equation must be done to the other in order to main-tain equality. For example: Add or subtract the same value to both sides. Multiply or divide both sides by the same value. Square or take the square root of both sides.
Algebra With Equations Formulas can be solved by per as long as ming a sequence of identical operations to both sides of an equality. Terms may be added or subtracted from each side of an equality. x = 2 – 4 + 6 x = +4 x + 4 – 6 = 2 (Example) Equations (Cont.) Each term on both sides can be multiplied or divided by the same factor. Equations (Cont.) Divide both sides by: (m2 m1)
Equations (Cont.) Equations (Cont.) Each side may be raised to a power or the root may be taken of each side. This is Getting Tougher! Man Arithmetic is one thing, but I gotta have help with solving as long as those letters.
Formula Rearrangement Consider the following as long as mula: Multiply by B to solve as long as A: Notice that B has moved up to the right. Thus, the solution as long as A becomes: Next Solve as long as D D moves up to left. A moves down to right. B moves up to right. D is then isolated. Cross Roads as long as Factors When there are only two terms in a as long as mula separated by an equals sign, the cross roads can be used. Cross Roads as long as Factors Only! Example solutions are given below:
Example 4: Solve as long as n. PV = nRT R T R T R T CAUTION SIGNS FOR CROSS ROADS The cross-road method works ONLY as long as FACTORS! The c cannot be moved unless the entire factor (b + c) is moved. Solution as long as a: Example 5: Solve as long as f. First move f to get it in numerator. Next move a, d, in addition to (b + c)
When to use Cross-Roads: 2. Only FACTORS may be moved! WARNING: DONT SHOW THIS CROSS ROADS APPROACH TO A MATH TEACHER! Use the technique because it works in addition to is effective. Recognize the problems of confusing factors with terms. BUT Dont expect all instructors to like it. Just use it quietly, in addition to dont tell anyone. Often It is Necessary to Use Exponents in Physics Applications. E = mc2
Trigonometry Review You are expected to know the following: y = R sin q x = R cos q R2 = x2 + y2 Trigonometry Conclusion of Chap. 2 Technical Mathematics
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