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.

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POLY ENGINEERING3-8

1. Energy Sources – Fuels and Power Plants2. Trigonometry and Vectors3. Classical Mechanics:

Force, Work, Energy, and Power4. Impacts of Current Generation and Use

UNIT 3 – ENERGY AND POWER

Topics Covered

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POLY ENGINEERING3-8

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

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POLY ENGINEERING3-8

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:

a2 + b2 = c2

Trigonometry

180

A

B

C

a, b, and c

a

b

c

HYPOTENUSE

A, B, and C

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POLY ENGINEERING3-8

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:

a2 + b2 = c2

Trigonometry

A

B

C

a

b

c

HYPOTENUSE

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)

3222 ba c

22 bac 169

5c

22 bac 22 11

2c

22 bac 22 1)3(

2c 13

Align equal signs when possible

Trigonometry and Vectors

Common triangles in Geometry and Trigonometry

3

4

5

1

Trigonometry and VectorsCommon triangles in Geometry and

Trigonometry

11

1

2

45o

45o

2

3

30o

60o

You must memorize these triangles

2 3

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 12a) b) c)

22 bca

36 6a

222 ba c 222 bc a

22 801

22 bca 22 2113

144169 25

5a

22 bca 22 2115

144225 81

9a

Divide all sides by 2 3-4-5 triangle

Divide all sides by 3 3-4-5 triangle

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POLY ENGINEERING3-8

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

OPP

OSI

TEADJACENT

sin A = ac

sin A = opposite

hypotenuse

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POLY ENGINEERING3-8

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

OPP

OSI

TEADJACENT

cos A = bc

cos A = adjacent

hypotenuse

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POLY ENGINEERING3-8

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

OPP

OSI

TEADJACENT

tan A = ab

tan A = opposite adjacent

Trigonometry and Vectors

3

4

51

2

3

1

1

2

NO CALCULATORS – SKETCH – SIMPLIFY ANSWERS

3. For <A below calculate Sine, Cosine, and Tangent:

Trigonometric Function Problems

A

B

C A

B

CA

B

C

a) b) c)

sin A = opp. hyp. cos A = adj.

hyp.tan A =

opp. adj.

Sketch and answer in your notebook

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 adjacentsin A = 3

5

cos A = 45

tan A = 34

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 adjacentsin A = 1

√2

cos A =

tan A = 1

1

1

2

A

B

C

b)

1 √2

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 adjacentsin A = 1

2

cos A =

tan A =

√3 2

12

3A

B

C

c)

1 √3

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POLY ENGINEERING3-8

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

Trigonometry and Vectors

Common triangles in Trigonometry

1

1

2

45o

45o

12

3

30o

60o

You must memorize these triangles

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POLY ENGINEERING3-8

Trigonometry and Vectors

Trigonometric FunctionsNO CALCULATORS – SKETCH – SIMPLIFY ANSWERS

4. Calculate sine, cosine, and tangent for the following angles:a. 30o

b. 60o

c. 45o

12

3

30o

60osin 30 =

12

cos 30 = √3 2

tan 30 = 1 √3

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POLY ENGINEERING3-8

Trigonometry and Vectors

Trigonometric FunctionsNO CALCULATORS – SKETCH – SIMPLIFY ANSWERS

4. Calculate sine, cosine, and tangent for the following angles:a. 30o

b. 60o

c. 45o

12

3

30o

60o

cos 60 = 12

sin 60 = √3 2

tan 60 = √3

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POLY ENGINEERING3-8

Trigonometry and Vectors

Trigonometric FunctionsNO CALCULATORS – SKETCH – SIMPLIFY ANSWERS

4. Calculate sine, cosine, and tangent for the following angles:a. 30o

b. 60o

c. 45o

tan 45 = 1

sin 45 = 1 √2

cos 45 = 1 √2

1

1

2

45o

45o

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POLY ENGINEERING3-8

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

0

90

180

270

Trigonometry and VectorsMeasuring 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

=

=

=

=

=

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.

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POLY ENGINEERING3-9

HOMEWORK

sin A = ac cos A =

bc

tan A = ab

45o

30o

45o

30o1 2 3

2

√2

√3 √3

√23 4

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POLY ENGINEERING3-9

HOMEWORK

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POLY ENGINEERING3-9

DRILLComplete #4 on the Trigonometry worksheet.

tan A = opposite adjacentsin 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

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 VectorsAlgebra Using Trig Functions

5 2a

sin a= y r

sin a= 2 5

x

We will now go over methods for solving #5 and #6 on

Trigonometry Worksheet

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POLY ENGINEERING3-9

Multiply both sides by rr ya

cos a= x r

r (cos a)= x 10Divide both sides by cos a

r = x

cos a Substitute and Solve= 10 2/5

= (10) 5 2

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 FunctionsTrigonometry and Vectors

Trigonometry and Vectors

HOMEWORK

1. Complete problems 4-6 on the Trig. Worksheet

[2. Will be covered shortly]

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POLY ENGINEERING3-9

Trigonometry and Vectors

1. Scalar Quantities – a quantity that involves magnitude only; direction is not importantTiger 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

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POLY ENGINEERING3-9

Trigonometry and Vectors

1. 5 miles northeast

2. 6 yards

3. 1000 lbs force

Scalar or Vector?

VectorMagnitude and Direction

ScalarMagnitude only

ScalarMagnitude only

4. 400 mph due north

5. $100

6. 10 lbs weight

VectorMagnitude and Direction

ScalarMagnitude only

VectorMagnitude and Direction

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POLY ENGINEERING3-9

Trigonometry and Vectors

3. Free-body DiagramA diagram that shows all external forces acting on an object.

Vectors

friction force

force of gravity

(weight)

applied force

normal force

Wt

FN

Ff

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POLY ENGINEERING3-9

Trigonometry and Vectors

4. Describing vectors – We MUST represent both magnitude and direction.

Describe the force applied to the wagon by the skeleton:

Vectors

45o40 lb

s

magnitude direction

F = 40 lbs 45o

Hat signifies vector quantity

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POLY ENGINEERING3-9

Trigonometry and Vectors

2 ways of describing vectors…

Vectors

45o40 lb

s

F = 40 lbs 45o

F = 40 lbs @ 45o

Students must use this form

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POLY ENGINEERING3-9

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

32

1

4 65

INITIAL SIDE

X” = Y lbs.

Zo

F = 3 13/16 lbs. < 14o

Answer to #1

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POLY ENGINEERING3-10

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

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POLY ENGINEERING3-10

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

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POLY ENGINEERING3-10

Trigonometry and VectorsVectors – Rectangular Components

y

x

F

Fx

Fy

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.

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POLY ENGINEERING3-10

Trigonometry and VectorsVectors – Rectangular Components

y

x

F

Fx

Fy

This means:

vector F = vector Fx + vector Fy

Remember the addition of vectors:

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POLY ENGINEERING3-10

Trigonometry and Vectors

Vectors – Rectangular Components

y

x

F

Fx

Fy

Fx = Fx i

Vector Fx = Magnitude Fx times vector i

Vector Fy = Magnitude Fy times vector j

Fy = Fy j

F = Fx i + Fy j

i denotes vector in x direction

j denotes vector in y direction

Unit vector

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POLY ENGINEERING3-10

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

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POLY ENGINEERING3-10

Trigonometry and Vectors

Vectors – Rectangular Components

y

x

F

Fx

Fy

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

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POLY ENGINEERING3-10

Trigonometry and Vectors

Vectors – Rectangular Components

F

Fx

Fy

cos Q = Fx / F

Fx = F cos Qi

sin Q = Fy / F

Fy = F sin Qj

What is the relationship between Q, sin Q, and cos Q?

Q

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POLY ENGINEERING3-10

Trigonometry and Vectors

Vectors – Rectangular Components

y

x

F Fx +

Fy +

When are Fx and Fy Positive/Negative?

FFx -

Fy +

FFFx -Fy -

Fx +Fy -

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POLY ENGINEERING3-10

Vectors – Rectangular Components

Complete the following chart in your notebook:

III

III IV

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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

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POLY ENGINEERING

Fx = F cos Qi Fy = F sin Qj

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POLY ENGINEERING

Fx = F cos Qi Fy = F sin Qj

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POLY ENGINEERING

Fx = F cos Qi Fy = F sin Qj

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POLY ENGINEERING3-10

Trigonometry and VectorsVectors – 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

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POLY ENGINEERING3-10

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 VectorsVectors – Resultant Forces

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

3

3

3

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POLY ENGINEERING3-10

Trigonometry and VectorsVectors – Resultant Forces

R = SFx + SFy

R = (75 lbs)i + (75 - 100 lbs)j

R = (75 lbs)i + (29.9 lbs)j

3

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

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POLY ENGINEERING3-13

WORK1. 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

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POLY ENGINEERING3-13

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.

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POLY ENGINEERING3-13

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.

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POLY ENGINEERING3-13

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?

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POLY ENGINEERING3-13

WORKWORK = 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 lbs W = F x dd = 1,450 ft. = (1.3 lb) x (1,450 ft.)W = ? W = 1,885 ft-lb

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POLY ENGINEERING3-13

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.

Wt = 3,000 lb

30o

Fw = ?d = 200’

WORKBack to our drill problem

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POLY ENGINEERING3-13

Wt = 3,000 lb

30o

Fw = ?d = 200’

WORK

60o

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POLY ENGINEERING3-13

WORK

60o

3000 lb.

x cos 60o = x / (3000 lb)

x = (3000 lb)(cos 600)

= (3000 lb)(1/2)

x = 1,500 lb.

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POLY ENGINEERING3-13

Wt = 3,000 lb

30o

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

EFFICIENCY

EFFICIENCY = x 100%OUTPUT INPUT

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POLY ENGINEERING3-13

Wt = 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

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POLY ENGINEERING3-13

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%

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POLY ENGINEERING3-13

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?

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POLY ENGINEERING3-13

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