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 UNIT 3 – ENERGY AND POWER Topics Covered
Jan 01, 2016
IOT
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
IOT
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
IOT
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
IOT
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
IOT
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
IOT
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
IOT
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
IOT
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
IOT
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
IOT
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
IOT
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
IOT
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.
IOT
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
IOT
POLY ENGINEERING3-9
HOMEWORK
IOT
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
IOT
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]
IOT
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
IOT
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
IOT
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
IOT
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
IOT
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
IOT
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
IOT
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
IOT
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
IOT
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.
IOT
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:
IOT
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
IOT
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
IOT
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
IOT
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
IOT
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 -
IOT
POLY ENGINEERING3-10
Vectors – Rectangular Components
Complete the following chart in your notebook:
III
III IV
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 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 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
IOT
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
IOT
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
IOT
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
IOT
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.
IOT
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.
IOT
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?
IOT
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
IOT
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
IOT
POLY ENGINEERING3-13
Wt = 3,000 lb
30o
Fw = ?d = 200’
WORK
60o
IOT
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.
IOT
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
IOT
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
IOT
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%
IOT
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?
IOT
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