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UNIT 3 – ENERGY AND POWER Topics Covered 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 IOT 3-8 POLY ENGINEERING Trigonometry and Vectors Background – Trigonometry 1. 2. 3. 4. 5. 6. 7. 8. Trigonometry, triangle measure, from Greek. Mathematics that deals with the sides and angles of triangles, and their relationships. Computational Geometry (Geometry – earth measure). Deals mostly with right triangles. Historically developed for astronomy and geography. Not the work of any one person or nation – spans 1000s yrs. REQUIRED for the study of Calculus. Currently used mainly in physics, engineering, and chemistry, with applications in natural and social sciences. IOT 3-8 POLY ENGINEERING Trigonometry and Vectors Trigonometry 1. 2. 3. 4. Total degrees in a triangle: 180 Three angles of the triangle below: A, B, and Three sides of the triangle below: a, b, and c Pythagorean Theorem: B C a2 + b2 = c2 c A b a C IOT 3-8 POLY ENGINEERING Trigonometry and Vectors Trigonometry 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: B a2 + b2 = c2 c A b a C IOT 3-8 POLY ENGINEERING Trigonometry and Vectors Trigonometry – Pyth. Thm. Problems NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 1. Solve for the unknown hypotenuse of the following triangles: a) b) ? 3 4 ? c) 1 ? 1 3 1 c2 a 2 b2 2 2 2 2 c a b c a b c a 2 b2 Align equal signs possible 2 2 2 when 2 ( 3) 1 1 1 9 16 3 1 c 2 c5 c2 Trigonometry and Vectors Common triangles in Geometry and Trigonometry 5 1 3 4 Trigonometry and Vectors Common triangles in Geometry and Trigonometry You must memorize these triangles 45o 60o 2 2 1 30o 45o 1 2 1 3 3 Trigonometry and Vectors Trigonometry – Pyth. Thm. Problems NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 2. Solve for the unknown side of the following triangles: a) 10 b) 8 ? c a b Divide all2 sides by 2 2 2 a c3-4-5 b triangle a c2 b2 2 2 2 102 82 36 13 a 6 ? c) 12 ? 12 15 a c2 b2 a c2 b2 132 122 169 144 25 a 5 Divide all2sides 15 122 by 3 3-4-5 triangle 225 144 81 a 9 Trigonometry and Vectors Trigonometric Functions – Sine Standard triangle labeling. Sine of <A is equal to the side opposite <A divided by the hypotenuse. opposite sin A = hypotenuse a sin A = c B c A ADJACENT b OPPOSITE 1. 2. a C IOT 3-8 POLY ENGINEERING Trigonometry and Vectors Trigonometric Functions – Cosine Standard triangle labeling. Cosine of <A is equal to the side adjacent <A divided by the hypotenuse. adjacent cos A = hypotenuse cos A = b c B c A ADJACENT b OPPOSITE 1. 2. a C IOT 3-8 POLY ENGINEERING Trigonometry and Vectors Trigonometric Functions – Tangent Standard triangle labeling. Tangent of <A is equal to the side opposite <A divided by the side adjacent <A. tan A = opposite adjacent tan A = a b B c A ADJACENT b OPPOSITE 1. 2. a C IOT 3-8 POLY ENGINEERING Trigonometry and Vectors Trigonometric Function Problems NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 3. For <A below calculate Sine, Cosine, and Tangent: B c) b) a) B 5 3 4 C 2 B 2 1 Sketch and answer in your notebook A opp. sin A = hyp. A 1 C tan A = opp. adj. A 3 cos A = 1 C adj. hyp. Trigonometry and Vectors Trigonometric Function Problems 3. For <A below, calculate Sine, Cosine, and Tangent: a) A B 5 4 opposite sin A = hypotenuse 3 C 3 sin A = 5 adjacent cos A = hypotenuse 4 cos A = 5 opposite tan A = adjacent 3 tan A = 4 Trigonometry and Vectors Trigonometric Function Problems 3. For <A below, calculate Sine, Cosine, and Tangent: B b) A 2 1 1 C opposite sin A = hypotenuse 1 sin A = √2 adjacent cos A = hypotenuse cos A = 1 √2 opposite tan A = adjacent tan A = 1 Trigonometry and Vectors Trigonometric Function Problems 3. For <A below, calculate Sine, Cosine, and Tangent: B c) A 2 1 3 C opposite sin A = hypotenuse 1 sin A = 2 opposite tan A = adjacent adjacent cos A = hypotenuse tan A = 1 √3 cos A = √3 2 Trigonometry and Vectors Trigonometric Functions Trigonometric functions are ratios of the lengths of the segments that make up angles. opposite sin A = hypotenuse adjacent cos A = hypotenuse tan A = opposite adjacent IOT 3-8 POLY ENGINEERING Trigonometry and Vectors Common triangles in Trigonometry You must memorize these triangles 45o 60o 2 2 1 1 30o 45o 1 3 Trigonometry and Vectors Trigonometric Functions NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 4. Calculate sine, cosine, and tangent for the following angles: a. 30o 1 b. 60o sin 30 = o 2 60 c. 45o 2 cos 30 = √3 2 tan 30 = 1 √3 1 30o 3 IOT 3-8 POLY ENGINEERING Trigonometry and Vectors Trigonometric Functions NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 4. Calculate sine, cosine, and tangent for the following angles: a. 30o √3 b. 60o sin 60 = o 2 60 c. 45o 2 1 cos 60 = 2 tan 60 = √3 1 30o 3 IOT 3-8 POLY ENGINEERING Trigonometry and Vectors Trigonometric Functions NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS 4. Calculate sine, cosine, and tangent for the following angles: a. 30o b. 60o 45o 1 cos 45 = o √2 c. 45 2 1 sin 45 = √2 tan 45 = 1 1 45o 1 IOT 3-8 POLY ENGINEERING Trigonometry and Vectors 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 30 degrees = -330 degrees 45 degrees = -315 degrees 90 degrees = -270 degrees 180 INITIAL SIDE 0 180 degrees = -180 degrees 270 degrees = -90 degrees IOT 270 360 degrees 3-8 POLY ENGINEERING 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. HOMEWORK a sin A = c cos A = tan A = o √2 45 1 2 30o √3 3√2 45o 3 b c a b 4 30o 2√3 IOT 3-9 POLY ENGINEERING HOMEWORK IOT 3-9 POLY ENGINEERING DRILL Complete #4 on the Trigonometry worksheet. opposite sin A = hypotenuse adjacent cos A = hypotenuse tan A = opposite adjacent sin = 3/16 sin = 5/16 sin = 1/2 sin = 5/8 tan = ~3/16 tan = 1/3 tan = 4/7 tan = 5/6 sin = 11/16 sin = 3/4 sin = 7/8 sin = 1/8 tan = 1 tan = 1 1/5 tan = 1 3/4 tan = ~1/8 IOT 3-9 POLY ENGINEERING 1. Sketch (sketches go on right side) 2.Trigonometry Write formula (andand alter Vectors if necessary) 3. Substitute and solve answers) Algebra Using Trig(box Functions 4. Check your solution (make sense?) We will now go over methods for solving #5 and #6 on y 2 sin a= Trigonometry r Worksheet a sin a= 2 5 x IOT 3-9 POLY ENGINEERING 1. Sketch (sketches go on right side) 2.Trigonometry Write formula (andand alter Vectors if necessary) 3. Substitute and solve answers) Algebra Using Trig(box Functions 4. Check your solution (make sense?) cos a= r (cos a)= Use to solve for y x r x Multiply both sides by r y a Divide both sides 10 by cos a x 10 r = cos a = 2/5 = (10) 5 2 r = 25 Substitute and Solve Trigonometry and Vectors HOMEWORK 1. Complete problems 4-6 on the Trig. Worksheet [2. Will be covered shortly] Trigonometry and Vectors 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 How hard to impact the cue ball is only part of the game – you need to know direction too Weight is a vector quantity IOT 3-9 POLY ENGINEERING Trigonometry and Vectors Scalar or Vector? 1. 5 miles northeast Magnitude and Direction Vector 2. 6 yards Magnitude only Scalar 3. 1000 lbs force Magnitude only Scalar 4. 400 mph due north Magnitude and Direction Vector 5. $100 Magnitude only Scalar 6. 10 lbs weight Magnitude and Direction Vector IOT 3-9 POLY ENGINEERING Trigonometry and Vectors Vectors 3. Free-body Diagram A diagram that shows all external forces acting on an object. normal force applied N force F Ff friction force Wt force of gravity (weight) IOT 3-9 POLY ENGINEERING Trigonometry and Vectors Vectors 4. Describing vectors – We MUST represent both magnitude and direction. Describe the force applied to the wagon by the skeleton: Hat signifies vector quantity 45o F = 40 lbs 45o IOT magnitude direction 3-9 POLY ENGINEERING Trigonometry and Vectors Vectors 2 ways of describing vectors… F = 40 lbs Students must use this form 45o F = 40 lbs @ 45o 45o IOT 3-9 POLY ENGINEERING Trigonometry and Vectors Describing 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. 3 2 1 Zo INITIAL SIDE Answer to #1 F = 3 13/16 lbs. < 14o 4 5 6 IOT 3-9 POLY ENGINEERING Trigonometry and Vectors Vectors – Scalar Multiplication 1. 2. We can multiply any vector by a whole number. Original direction is maintained, new magnitude. 2 ½ IOT 3-10 POLY ENGINEERING Trigonometry and Vectors Vectors – Addition 1. 2. 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) IOT 3-10 POLY ENGINEERING Trigonometry and Vectors Vectors – Rectangular Components 1. 2. 3. 4. 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 Fy IOT Fx x 3-10 POLY ENGINEERING Trigonometry and Vectors Vectors – Rectangular Components This means: vector F = vector Fx + vector Fy Remember the addition of vectors: y Fy Fx x IOT 3-10 POLY ENGINEERING Trigonometry and Vectors Unit vector Vectors – Rectangular Components Vector Fx = Magnitude Fx times vector i F = Fx i + Fy j Fx = Fx i i denotes vector in x direction y Vector Fy = Magnitude Fy times vector j Fy = Fy j Fy j denotes vector in y direction Fx x IOT 3-10 POLY ENGINEERING Trigonometry and Vectors 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 IOT 3-10 POLY ENGINEERING Trigonometry and Vectors Vectors – Rectangular Components Each grid space represents 1 lb force. What is Fx? y Fx = (4 lbs)i What is Fy? Fy Fy = (3 lbs)j Fx x What is F? F = (4 lbs)i + (3 lbs)j IOT 3-10 POLY ENGINEERING Trigonometry and Vectors Vectors – Rectangular Components What is the relationship between Q, sin Q, and cos Q? cos Q = Fx / F Fx = F cos Qi Fy sin Q = Fy / F Fy = F sin Qj Q Fx IOT 3-10 POLY ENGINEERING Trigonometry and Vectors Vectors – Rectangular Components When are Fx and Fy Positive/Negative? Fy + y Fy + Fx + Fx x Fx Fy - Fy - Fx + IOT 3-10 POLY ENGINEERING Vectors – Rectangular Components Complete the following chart in your notebook: II I III IV IOT 3-10 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 Fx = F cos Qi Fy = F sin Qj IOT POLY ENGINEERING Fx = F cos Qi Fy = F sin Qj IOT POLY ENGINEERING Fx = F cos Qi Fy = F sin Qj IOT POLY ENGINEERING Fx = F cos Qi Fy = F sin Qj IOT POLY ENGINEERING 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 Fx = F cos Qi = (150 lbs) (cos 60) i No x-component = (75 lbs)i SFx = (75 lbs)i IOT 3-10 POLY ENGINEERING 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 Fy = F sin Qj = (150 lbs) (sin 60) j = (75 3 lbs)j Wy = -(100 lbs)j SFy = (75 3 lbs)j - (100 lbs)j SFy = (75 3 - 100 lbs)j IOT 3-10 POLY ENGINEERING 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 R = SFx + SFy R = (75 lbs)i + (75 3 - 100 lbs)j R = (75 lbs)i + (29.9 lbs)j IOT 3-10 POLY ENGINEERING 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 IOT 3-13 POLY ENGINEERING 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. IOT 3-13 POLY ENGINEERING 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. IOT 3-13 POLY ENGINEERING WORK WORK = FORCE x DISTANCE W=Fxd Consider the 1.3-lb ball below, sitting at rest. How much work is gravity doing on the ball? IOT 3-13 POLY ENGINEERING WORK WORK = FORCE x DISTANCE W=Fxd 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 lbs d = 1,450 ft. W=? W=Fxd = (1.3 lb) x (1,450 ft.) W = 1,885 ft-lb IOT 3-13 POLY ENGINEERING WORK Back to our drill problem A 3,000-lb car is sitting on a hill in neutral. The angle the hill makes with the horizontal is 30o. 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. Fw = ? Wt = 3,000 lb 30o IOT 3-13 POLY ENGINEERING WORK Fw = ? 60o Wt = 3,000 lb 30o IOT 3-13 POLY ENGINEERING WORK x 60o cos 60o = x / (3000 lb) x = (3000 lb)(cos 600) 3000 lb. = (3000 lb)(1/2) x = 1,500 lb. IOT 3-13 POLY ENGINEERING WORK F = 1,500 lb F = 1,500 lb. d = 200 ft Wt = 3,000 lb W=? 30o W = Fxd = (1500 lb) x (200 ft) W = 300,000 ft-lb IOT 3-13 POLY ENGINEERING EFFICIENCY OUTPUT EFFICIENCY = x 100% INPUT EFFICIENCY Back to our drill problem F = 1,500 lb. Wt = 3,000 lb FORCE APPLIED = 3,000 lb INPUT EFFECTIVE FORCE = 1,500 lb OUTPUT IOT 3-13 POLY ENGINEERING EFFICIENCY Back to our drill problem FORCE APPLIED = 3,000 lb INPUT EFFECTIVE FORCE = 1,500 lb OUTPUT EFFICIENCY = EFF = OUTPUT x 100% INPUT 1,500 lb x 100% 3,000 lb EFF = 50% IOT 3-13 POLY ENGINEERING POWER 1. 2. 3. 4. 5. Three Buddhist monks walk up stairs to a temple. Each weighs 150 lbs and climbs height of 100’. One climbs faster than the other two. Who does more work? 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? IOT 3-13 POLY ENGINEERING POWER Power is the rate of doing Work W P= t The less time it takes…. The more power Units: Watts, Horsepower, Ft-lbs/s IOT 3-13 POLY ENGINEERING