IOT POLY ENGINEERING

1. IOTPOLYENGINEERING 3-10 January __, 2009 DRILL • With a partner, go over your solutions to last night’s homework. Make sure all work is neat and any incongruence between answers is resolved. • Last night’s homework: • Complete problems 4-6 on the Trig. Worksheet • 2. Complete problems 1-2 on the Vector Worksheet

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6. IOTPOLYENGINEERING 3-9

7. Trigonometry and Vectors IOTPOLYENGINEERING 3-8 REVIEW Trigonometry Pythagorean Theorem: r2 = x2 + y2 B r y HYPOTENUSE A C x

8. Trigonometry and Vectors REVIEW Common triangles in Geometry and Trigonometry 5 3 4 1

9. Trigonometry and Vectors REVIEW Common triangles in Geometry and Trigonometry You must memorize these triangles 45o 60o 2 1 1 30o 45o 1 2 3

10. Trigonometry and Vectors opposite hypotenuse sin A = opposite adjacent adjacent hypotenuse tan A = cos A = IOTPOLYENGINEERING 3-8 REVIEW Trigonometric Functions Trigonometric functions are ratios of the lengths of the segments that make up angles.

11. Trigonometry and Vectors REVIEW Measuring Angles • Unless otherwise specified: • Positive angles measured counter-clockwise from the horizontal. • Negative angles measured clockwise from the horizontal. • We call the horizontal line 0o, or the initial side 90 -330 degrees -315 degrees -270 degrees -180 degrees -90 degrees 30 degrees 45 degrees 90 degrees 180 degrees 270 degrees 360 degrees = = = = = 180 0 INITIAL SIDE IOTPOLYENGINEERING 3-8 270

12. Trigonometry and Vectors IOTPOLYENGINEERING 3-9 REVIEW Vectors • Scalar Quantities – a quantity that involves magnitude only; direction is not important • Tiger Woods – 6’1” • Shaquille O’Neill – 7’0” • Vector Quantities – a quantity that involves both magnitudeand direction How hard to impact the cue ball is only part of the game – you need to know direction too Weight is a vector quantity

13. Trigonometry and Vectors IOTPOLYENGINEERING 3-9 REVIEW Scalar or Vector? • 400 mph due north • $100 • 10 lbs weight • 5 miles northeast • 6 yards • 1000 lbs force Magnitude and Direction Magnitude and Direction Vector Vector Magnitude only Magnitude only Scalar Scalar Magnitude and Direction Magnitude only Vector Scalar

14. Trigonometry and Vectors IOTPOLYENGINEERING 3-9 REVIEW Vectors • Free-body Diagram • A diagram that shows all external forces acting on an object. applied force normal force N F Ff friction force force of gravity (weight) Wt

15. Trigonometry and Vectors IOTPOLYENGINEERING 3-9 REVIEW Vectors • Describing vectors – • We MUST represent both magnitudeand direction. • Describe the force applied to the wagon by the skeleton: Hat signifies vector quantity 40 lbs 45o F = 40 lbs 45o magnitude direction

16. Trigonometry and Vectors IOTPOLYENGINEERING 3-9 REVIEW Vectors • 2 ways of describing vectors… Students must use this form F = 40 lbs 45o F = 40 lbs @ 45o 40 lbs 45o

17. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Scalar Multiplication • We can multiply any vector by a whole number. • Original direction is maintained, new magnitude. 2 ½

18. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Addition • We can add two or more vectors together. • Redraw vectors head-to-tail, then draw the resultant vector. • (head-to-tail order does not matter)

19. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Rectangular Components • It is often useful to break a vector into horizontal and vertical components (rectangular components). • Consider the Force vector below. • Plot this vector on x-y axis. • Project the vector onto x and y axes. y F Fy x Fx

20. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Rectangular Components This means: vector F = vector Fx + vector Fy Remember the addition of vectors: y F Fy x Fx

21. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Unit vector Vectors – Rectangular Components Vector Fx= Magnitude Fxtimes vector i F = Fx i + Fyj Fx= Fx i i denotes vector in x direction y Vector Fy= Magnitude Fytimes vector j F Fy= Fy j Fy j denotes vector in y direction x Fx

22. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Rectangular Components From now on, vectors on this screen will appear as bold type without hats. For example, Fx = (4 lbs)i Fy = (3 lbs)j F = (4 lbs)i + (3 lbs)j

23. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Rectangular Components Each grid space represents 1 lb force. What is Fx? Fx = (4 lbs)i What is Fy? Fy = (3 lbs)j What is F? F = (4 lbs)i + (3 lbs)j y F Fy x Fx

24. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Rectangular Components What is the relationship between Q, sin Q, and cos Q? cos Q = Fx / F Fx = F cos Qi sin Q = Fy / F Fy = F sin Qj F Fy Q Fx

25. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Rectangular Components When are Fx and Fy Positive/Negative? Fy + Fy + y F Fx + Fx - F x F F Fx - Fx + Fy - Fy -

26. IOTPOLYENGINEERING 3-10 Vectors – Rectangular Components Complete the following chart in your notebook: I II III IV

27. Rewriting vectors in terms of rectangular components: 1) Find force in x-direction – write formula and substitute 2) Find force in y-direction – write formula and substitute 3) Write as a single vector in rectangular components IOTPOLYENGINEERING Fx = F cos Qi Fy = F sin Qj

28. IOTPOLYENGINEERING Fx = F cos Qi Fy = F sin Qj

29. IOTPOLYENGINEERING Fx = F cos Qi Fy = F sin Qj

30. IOTPOLYENGINEERING Fx = F cos Qi Fy = F sin Qj

31. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Resultant Forces Resultant forces are the overall combination of all forces acting on a body. 1) sum of forces in x-direction 2) sum of forces in y-direction 3) Write as single vector in rectangular components Fx = F cos Qi = (150 lbs) (cos 60) i = (75 lbs)i SFx= (75 lbs)i No x-component

32. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Resultant Forces Resultant forces are the overall combination of all forces acting on a body. 1) sum of forces in x-direction 2) sum of forces in y-direction 3) Write as single vector in rectangular components Fy = F sin Qj = (150 lbs) (sin 60) j = (75 lbs)j Wy = -(100 lbs)j SFy= (75 lbs)j - (100 lbs)j SFy = (75 - 100 lbs)j

33. Trigonometry and Vectors IOTPOLYENGINEERING 3-10 Vectors – Resultant Forces Resultant forces are the overall combination of all forces acting on a body. 1) sum of forces in x-direction 2) sum of forces in y-direction 3) Write as single vector in rectangular components R = SFx +SFy R = (75 lbs)i + (75 - 100 lbs)j R = (75 lbs)i + (29.9 lbs)j

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35. IOTPOLYENGINEERING 3-10 Trigonometry and Vectors

36. IOTPOLYENGINEERING 3-10 Trigonometry and Vectors

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39. IOTPOLYENGINEERING 3-10 Trigonometry and Vectors

40. IOTPOLYENGINEERING 3-10 Trigonometry and Vectors CLASSWORK/ HOMEWORK Complete problem #4 on the Vector Worksheet