1. IOT POLY ENGINEERING 3-8 1.Energy Sources – Fuels and Power Plants 2.Trigonometry and Vectors 3.Classical Mechanics: Force, Work, Energy, and Power 4.Impacts of Current Generation and Use U NIT 3 – E NERGY AND P OWER Topics Covered

2. IOT POLY ENGINEERING 3-8 Trigonometry and Vectors 1.Trigonometry, triangle measure, from Greek. 2.Mathematics that deals with the sides and angles of triangles, and their relationships. 3.Computational Geometry (Geometry – earth measure). 4.Deals mostly with right triangles. 5.Historically developed for astronomy and geography. 6.Not the work of any one person or nation – spans 1000s yrs. 7.REQUIRED for the study of Calculus. 8.Currently used mainly in physics, engineering, and chemistry, with applications in natural and social sciences. Background – Trigonometry

3. IOT POLY ENGINEERING 3-8 Trigonometry and Vectors 1.Total degrees in a triangle: 2.Three angles of the triangle below: 3.Three sides of the triangle below: 4.Pythagorean Theorem: a 2 + b 2 = c 2 Trigonometry 180 A B C a, b, and c a b c HYPOTENUSE A, B, and C

4. IOT POLY ENGINEERING 3-8 Trigonometry and Vectors State the Pythagorean Theorem in words: “The sum of the squares of the two sides of a right triangle is equal to the square of the hypotenuse.” Pythagorean Theorem: a 2 + b 2 = c 2 Trigonometry A B C a b c HYPOTENUSE

5. Trigonometry and Vectors NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 1.Solve for the unknown hypotenuse of the following triangles: Trigonometry – Pyth. Thm. Problems 4 3 ? a) 1 1 ? b) 1 ? c) Align equal signs when possible

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

7. 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

8. Trigonometry and Vectors NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 2.Solve for the unknown side of the following triangles: Trigonometry – Pyth. Thm. Problems 8 ? 10 ? 15 ? 12 13 12 a) b) c) Divide all sides by 2 3-4-5 triangle Divide all sides by 3 3-4-5 triangle

9. IOT POLY ENGINEERING 3-8 Trigonometry and Vectors 1.Standard triangle labeling. 2.Sine of <A is equal to the side opposite <A divided by the hypotenuse. Trigonometric Functions – Sine A B C a b c HYPOTENUSE OPPOSITE ADJACENT sin A = acac opposite hypotenuse

10. IOT POLY ENGINEERING 3-8 Trigonometry and Vectors 1.Standard triangle labeling. 2.Cosine of <A is equal to the side adjacent <A divided by the hypotenuse. Trigonometric Functions – Cosine A B C a b c HYPOTENUSE OPPOSITE ADJACENT cos A = bcbc adjacent hypotenuse

11. IOT POLY ENGINEERING 3-8 Trigonometry and Vectors 1.Standard triangle labeling. 2.Tangent of <A is equal to the side opposite <A divided by the side adjacent <A. Trigonometric Functions – Tangent A B C a b c HYPOTENUSE OPPOSITE ADJACENT tan A = abab opposite adjacent

12. Trigonometry and Vectors 3 4 5 1 2 1 1 NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 3.For <A below calculate Sine, Cosine, and Tangent: Trigonometric Function Problems A B C A B C A B C a) b) c) sin A = opp. hyp. cos A = adj. hyp. tan A = opp. adj. Sketch and answer in your notebook

13. Trigonometry and Vectors 3 4 5 3.For <A below, calculate Sine, Cosine, and Tangent: Trigonometric Function Problems A B C a) sin A = opposite hypotenuse cos A = adjacent hypotenuse tan A = opposite adjacent sin A = 3535 cos A = 4545 tan A = 3434

14. Trigonometry and Vectors 3.For <A below, calculate Sine, Cosine, and Tangent: Trigonometric Function Problems sin A = opposite hypotenuse cos A = adjacent hypotenuse tan A = opposite adjacent sin A = 1 √2 cos A = tan A = 1 1 1 A B C b) 1 √2

15. Trigonometry and Vectors 3.For <A below, calculate Sine, Cosine, and Tangent: Trigonometric Function Problems sin A = opposite hypotenuse cos A = adjacent hypotenuse tan A = opposite adjacent sin A = 1212 cos A = tan A = √3 2 1 2 A B C c) 1 √3

16. 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

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

18. IOT POLY ENGINEERING 3-8 Trigonometry and Vectors Trigonometric Functions NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 4.Calculate sine, cosine, and tangent for the following angles: a.30 o b.60 o c.45 o 1 2 30 o 60 o sin 30 = 1212 cos 30 = √3 2 tan 30 = 1 √3

19. IOT POLY ENGINEERING 3-8 Trigonometry and Vectors Trigonometric Functions NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 4.Calculate sine, cosine, and tangent for the following angles: a.30 o b.60 o c.45 o 1 2 30 o 60 o cos 60 = 1212 sin 60 = √3 2 tan 60 = √3

20. IOT POLY ENGINEERING 3-8 Trigonometry and Vectors Trigonometric Functions NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 4.Calculate sine, cosine, and tangent for the following angles: a.30 o b.60 o c.45 o tan 45 = 1 sin 45 = 1 √2 cos 45 = 1 √2 1 1 45 o

21. 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 ==========

22. Trigonometry and Vectors Begin all lines as light construction lines! Draw the initial side – horizontal line. From each vertex, precisely measure the angle with a protractor. Measure 1” along the hypotenuse. Using protractor, draw vertical line from the 1” point. Darken the triangle.

23. IOT POLY ENGINEERING 3-9 HOMEWORK sin A = acac cos A = bcbc tan A = abab 45 o 30 o 45 o 30 o 1 2 3 2 √2 √3 √2 3 4

24. IOT POLY ENGINEERING 3-9 HOMEWORK

25. IOT POLY ENGINEERING 3-9 DRILL Complete #4 on the Trigonometry worksheet. tan A = opposite adjacent sin A = opposite hypotenuse cos A = adjacent hypotenuse sin = 3/16 tan = ~3/16 sin = 5/16 tan = 1/3 sin = 1/2 tan = 4/7 sin = 5/8 tan = 5/6 sin = 11/16 tan = 1 sin = 3/4 tan = 1 1/5 sin = 7/8 tan = 1 3/4 sin = 1/8 tan = ~1/8

26. 1.Sketch (sketches go on right side) 2.Write formula (and alter if necessary) 3.Substitute and solve (box answers) 4.Check your solution (make sense?) Trigonometry and Vectors Algebra Using Trig Functions 5 2 a sin a = yryr 2525 x We will now go over methods for solving #5 and #6 on Trigonometry Worksheet

27. IOT POLY ENGINEERING 3-9

28. Multiply both sides by r r y a cos a = xrxr r (cos a) = x 10 Divide both sides by cos a r = x cos a Substitute and Solve = 10 2/5 = (10) 5252 r = 25 25 Use to solve for y 1.Sketch (sketches go on right side) 2.Write formula (and alter if necessary) 3.Substitute and solve (box answers) 4.Check your solution (make sense?) Algebra Using Trig Functions Trigonometry and Vectors

30. HOMEWORK 1.Complete problems 4-6 on the Trig. Worksheet [2. Will be covered shortly]

31. 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

32. 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

33. 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

34. 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

35. 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

36. IOT POLY ENGINEERING 3-9 Trigonometry and Vectors Describe the force needed to shoot the cue ball into each pocket: Draw a line from center of cue ball to center of pocket. Measure the length of line: 1” = 1 lb force. Measure the required angle from the given initial side. Describing Vectors 3 2 1 4 6 5 INITIAL SIDE X” = Y lbs. ZoZo F = 3 13/16 lbs. < 14 o Answer to #1

37. 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

38. 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

39. 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.

40. 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:

41. 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

42. 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

43. 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

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

45. 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 -

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

47. 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

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

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

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

51. 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

52. 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

53. 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

54. IOT POLY ENGINEERING 3-13 WORK 1.Velocity, acceleration, force, etc. mean nearly the same thing in everyday life as they do in physics. 2.Work means something distinctly different. 3.Consider the following: 1)Hold a book at arm’s length for three minutes. 2)Your arm gets tired. 3)Did you do work? 4)No, you did no work whatsoever. 4.You exerted a force to support the book, but you did not move it. 5.A force does no work if the object doesn’t move

55. IOT POLY ENGINEERING 3-13 WORK The man below is holding 1 ton above his head. Is he doing work? No, the object is not moving. Describe the work he did do: Lifting the 1 ton from the ground to above his head.

56. IOT POLY ENGINEERING 3-13 WORK WORK = FORCE x DISTANCE The work W done on an object by an agent exerting a constant force on the object is the product of the component of the force in the direction of the displacement and the magnitude of the displacement.

57. IOT POLY ENGINEERING 3-13 WORK WORK = FORCE x DISTANCE W = F x d Consider the 1.3-lb ball below, sitting at rest. How much work is gravity doing on the ball?

58. IOT POLY ENGINEERING 3-13 WORK WORK = FORCE x DISTANCE W = F x d Now consider the 1.3-lb ball below, falling 1,450 ft from the top of Sears Tower. How much work will have gravity done on the ball by the time it hits the ground? F = 1.3 lbsW = F x d d = 1,450 ft. = (1.3 lb) x (1,450 ft.) W = ?W = 1,885 ft-lb

59. IOT POLY ENGINEERING 3-13 A 3,000-lb car is sitting on a hill in neutral. The angle the hill makes with the horizontal is 30 o. The distance from flat ground to the car is 200 ft. Begin with a free-body diagram. Then, calculate the weight component facing down the hill. Finally, calculate the work done on the car by gravity. W t = 3,000 lb 30 o F w = ? d = 200’ WORK Back to our drill problem

60. IOT POLY ENGINEERING 3-13 W t = 3,000 lb 30 o F w = ? d = 200’ WORK 60 o

61. IOT POLY ENGINEERING 3-13 WORK 60 o 3000 lb. x cos 60 o = x / (3000 lb) x = (3000 lb)(cos 60 0 ) = (3000 lb)(1/2) x = 1,500 lb.

62. IOT POLY ENGINEERING 3-13 W t = 3,000 lb 30 o F = 1,500 lb. d = 200’ WORK F = 1,500 lb d = 200 ft W = ? W = F x d = (1500 lb) x (200 ft) W = 300,000 ft-lb

63. EFFICIENCY

65. EFFICIENCY = x 100% OUTPUT INPUT

66. IOT POLY ENGINEERING 3-13 W t = 3,000 lb F = 1,500 lb. EFFICIENCY FORCE APPLIED = 3,000 lb EFFECTIVE FORCE = 1,500 lb Back to our drill problem INPUT OUTPUT

67. IOT POLY ENGINEERING 3-13 EFFICIENCY FORCE APPLIED = 3,000 lb EFFECTIVE FORCE = 1,500 lb Back to our drill problem INPUT OUTPUT EFFICIENCY = x 100% OUTPUT INPUT EFF = x 100% 1,500 lb 3,000 lb EFF =50%

68. IOT POLY ENGINEERING 3-13 POWER 1.Three Buddhist monks walk up stairs to a temple. 2.Each weighs 150 lbs and climbs height of 100’. 3.One climbs faster than the other two. 4.Who does more work? 5.They all do the same work: W = F x d(force for all three is 150 lb) = (150 lb)(100’) W = 15,000 ft-lb 6.Who has greater power?

69. IOT POLY ENGINEERING 3-13 POWER Power is the rate of doing Work P = The less time it takes…. The more power Units: Watts, Horsepower, Ft-lbs/s W t