1. IOT POLY ENGINEERING 3-10 DRILL January __, 2009 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: 1.Complete problems 4-6 on the Trig. Worksheet 2. Complete problems 1-2 on the Vector Worksheet

5. IOT POLY ENGINEERING 3-9 N

6. IOT POLY ENGINEERING 3-9

7. IOT POLY ENGINEERING 3-8 Trigonometry and Vectors Pythagorean Theorem: r 2 = x 2 + y 2 Trigonometry A B C y x r HYPOTENUSE

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

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

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

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

12. IOT POLY ENGINEERING 3-9 Trigonometry and Vectors 1.Scalar Quantities – a quantity that involves magnitude only; direction is not important Tiger Woods – 6’1” Shaquille O’Neill – 7’0” 2.Vector Quantities – a quantity that involves both magnitude and direction Vectors 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. IOT POLY ENGINEERING 3-9 Trigonometry and Vectors 1.5 miles northeast 2.6 yards 3.1000 lbs force Scalar or Vector? Vector Magnitude and Direction Scalar Magnitude only Scalar Magnitude only 4.400 mph due north 5.$100 6.10 lbs weight Vector Magnitude and Direction Scalar Magnitude only Vector Magnitude and Direction

14. IOT POLY ENGINEERING 3-9 Trigonometry and Vectors 3.Free-body Diagram A diagram that shows all external forces acting on an object. Vectors friction force force of gravity (weight) applied force normal force WtWt F N FfFf

15. IOT POLY ENGINEERING 3-9 Trigonometry and Vectors 4.Describing vectors – We MUST represent both magnitude and direction. Describe the force applied to the wagon by the skeleton: Vectors 45 o 40 lbs magnitude direction F = 40 lbs 45 o Hat signifies vector quantity

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

17. IOT POLY ENGINEERING 3-10 Trigonometry and Vectors 1.We can multiply any vector by a whole number. 2.Original direction is maintained, new magnitude. Vectors – Scalar Multiplication 2 ½

18. IOT POLY ENGINEERING 3-10 Trigonometry and Vectors 1.We can add two or more vectors together. 2.Redraw vectors head-to-tail, then draw the resultant vector. (head-to-tail order does not matter) Vectors – Addition

19. IOT POLY ENGINEERING 3-10 March 14, 2010 Drill: Draw these vectors Find 2 a and a +b y x b a a

20. IOT POLY ENGINEERING 3-10 a 2 a

21. IOT POLY ENGINEERING 3-10 y x b a a a+b

22. IOT POLY ENGINEERING 3-10 y x b a b a+b

23. IOT POLY ENGINEERING 3-10 Trigonometry and Vectors Vectors – Rectangular Components y x F FxFx FyFy 1.It is often useful to break a vector into horizontal and vertical components (rectangular components). 2.Consider the Force vector below. 3.Plot this vector on x-y axis. 4.Project the vector onto x and y axes.

24. IOT POLY ENGINEERING 3-10 Trigonometry and Vectors Vectors – Rectangular Components y x F FxFx FyFy This means: vector F = vector F x + vector F y Remember the addition of vectors:

25. IOT POLY ENGINEERING 3-10 Trigonometry and Vectors Vectors – Rectangular Components y x F FxFx FyFy F x = F x i Vector F x = Magnitude F x times vector i Vector F y = Magnitude F y times vector j F y = F y j F = F x i + F y j i denotes vector in x direction j denotes vector in y direction Unit vector

26. IOT POLY ENGINEERING 3-10 Trigonometry and Vectors Vectors – Rectangular Components From now on, vectors on this screen will appear as bold type without hats. For example, F x = (4 lbs)i F y = (3 lbs)j F = (4 lbs)i + (3 lbs)j

27. IOT POLY ENGINEERING 3-10 Trigonometry and Vectors Vectors – Rectangular Components y x F FxFx FyFy Each grid space represents 1 lb force. What is F x ? F x = (4 lbs)i What is F y ? F y = (3 lbs)j What is F? F = (4 lbs)i + (3 lbs)j

28. IOT POLY ENGINEERING 3-10 Trigonometry and Vectors Vectors – Rectangular Components F FxFx FyFy cos  = F x / F F x = F cos  i sin  = F y / F F y = F sin  j What is the relationship between , sin , and cos  ? 

29. IOT POLY ENGINEERING 3-10 Trigonometry and Vectors Vectors – Rectangular Components y x F F x + F y + When are F x and F y Positive/Negative? F F x - F y + F F F x - F y - F x + F y -

30. IOT POLY ENGINEERING 3-10 Vectors – Rectangular Components Complete the following chart in your notebook: I II III IV

31. Each grid space represents 1 lb force. What is F x ? F x = (5 lbs)i What is F y ? F y = (2 lbs)j What is F? F = (5 lbs)i + (2 lbs)j IOT POLY ENGINEERING 3-10 Trigonometry and Vectors Vectors – Rectangular Components y x F FxFx FyFy

32. IOT POLY ENGINEERING 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 F x = F cos Q i F y = F sin Q j

33. IOT POLY ENGINEERING F x = F cos Q i F y = F sin Q j

34. IOT POLY ENGINEERING F x = F cos Q i F y = F sin Q j

35. IOT POLY ENGINEERING F x = F cos Q i F y = F sin Q j

36. IOT POLY ENGINEERING 3-10 Trigonometry and Vectors 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 F x = F cos Q i = (150 lbs) (cos 60) i = (75 lbs)i S F x = (75 lbs)i No x-component

37. IOT POLY ENGINEERING 3-10 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 Trigonometry and Vectors Vectors – Resultant Forces F y = F sin Q j = (150 lbs) (sin 60) j = (75 lbs)j W y = -(100 lbs)j S F y = (75 lbs)j - (100 lbs)j S Fy = (75 - 100 lbs)j

38. IOT POLY ENGINEERING 3-10 Trigonometry and Vectors Vectors – Resultant Forces R = S F x + S F y R = (75 lbs)i + (75 - 100 lbs)j R = (75 lbs)i + (29.9 lbs)j 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

39. Trigonometry and Vectors IOT POLY ENGINEERING 3-10

40. Trigonometry and Vectors IOT POLY ENGINEERING 3-10

41. Trigonometry and Vectors IOT POLY ENGINEERING 3-10

42. Trigonometry and Vectors IOT POLY ENGINEERING 3-10

43. IOT POLY ENGINEERING 3-10 Trigonometry and Vectors Vectors – Rectangular Components F FxFx FyFy cos  = F x / F F x = F cos  i sin  = F y / F F y = F sin  j What is the relationship between , sin , and cos  ? 

44. Problem 4a IOT POLY ENGINEERING 3-10 100lbs 60 150lbs Gravity Space Junk Gravity: Space Junk:

45. Problem 4b IOT POLY ENGINEERING 3-10 28 lbs 5 lb 15lbs Gravity Gravity: Friction: Pulling Force 30 o Friction Pulling Force

46. Problem 4c IOT POLY ENGINEERING 3-10 1 lbs 4 lb 110lbs Gravity Gravity: Wind: Kick 45 o Wind Kick

47. Problem 4d IOT POLY ENGINEERING 3-10 3500 lbs 55 lbs 800lbs Gravity Gravity: Drag: Car Drag Car:

48. Trigonometry and Vectors IOT POLY ENGINEERING 3-10

49. Trigonometry and Vectors IOT POLY ENGINEERING 3-10

50. Trigonometry and Vectors CLASSWORK / HOMEWORK Complete problem #4 on the Vector Worksheet IOT POLY ENGINEERING 3-10