Download 3-18 IOT - Review 4 - Trig_ Vectors_ Work

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
c5
c2
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