January __, 2009
DRILL
With a partner, go over your solutions to last night’s
homework. Make sure all work is neat and any
incongruence between answers is resolved.
Last night’s homework:
1. Complete problems 4-6 on the Trig. Worksheet
2. Complete problems 1-2 on the Vector Worksheet
IOT
3-10
POLY ENGINEERING
N
IOT
3-9
POLY ENGINEERING
IOT
3-9
POLY ENGINEERING
Trigonometry and Vectors
Trigonometry
Pythagorean Theorem:
r2 = x2 + y2
B
r
A
y
x
C
IOT
3-8
POLY ENGINEERING
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
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
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
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
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
March 14, 2010
Drill: Draw these vectors
Find 2 a and a +b
y
a
a
b
x
IOT
3-10
POLY ENGINEERING
2a
a
IOT
3-10
POLY ENGINEERING
y
a
a+b
a
b
x
IOT
3-10
POLY ENGINEERING
y
b
a
a+b
b
x
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 q i
Fy
q
sin q = Fy / F
Fy = F sin q j
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
Trigonometry and Vectors
Vectors – Rectangular Components
Each grid space represents 1 lb force.
What is Fx?
y
Fx = (5 lbs)i
What is Fy?
Fy
Fy = (2 lbs)j
Fx
x
What is F?
F = (5 lbs)i + (2 lbs)j
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
Trigonometry and Vectors
IOT
3-10
POLY ENGINEERING
Trigonometry and Vectors
IOT
3-10
POLY ENGINEERING
Trigonometry and Vectors
IOT
3-10
POLY ENGINEERING
Trigonometry and Vectors
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 q i
Fy
q
sin q = Fy / F
Fy = F sin q j
Fx
IOT
3-10
POLY ENGINEERING
Problem 4a
60
150lbs
100lbs
Space
Junk
Gravity:
Space
Junk:
Gravity
Fgravity  0i  100 j
Fspace junk  150 cos60i  150 sin 60 j
Fres  Fgravity  FSpace Junk
Fres  75lbs i  230lbs  j
1
3
Fspace junk  150 * i  150 *
j
2
2
Fspace junk  75i  130 j
IOT
3-10
POLY ENGINEERING
Problem 4b
15lbs
5 lb
30o
Pulling Force
Friction
28 lbs
Gravity
Gravity:
Fgravity  0i  28 j
Friction:
F friction  5i  0 j
Pulling
Force
Fres  Fgravity  FFriction  FPulling
Fres  8lbs i  20.5lbs  j
FPulling  15 cos 30i  15 sin 30 j
FPulling  13i  7.5 j
IOT
3-10
POLY ENGINEERING
Problem 4c
110lbs
4 lb
45o
Kick
Wind
1 lbs
Gravity
Gravity:
Fgravity  0i  1 j
Wind:
FW ind  4i  0 j
Kick
Fres  Fgravity  FW ind  FKick
Fres  73.8lbs i  76.8lbs  j
FKick  110 cos 45i  110 sin 45 j
FPulling  77.8i  77.8 j
IOT
3-10
POLY ENGINEERING
Problem 4d
55 lbs
800lbs
Drag
Car
3500 lbs
Gravity
Gravity:
Fgravity  0i  3500 j
Drag:
FDrag  55i  0 j
Car:
FCar  800i  0 j
Fres  Fgravity  FDrag  FCar
Fres  745lbs i  3500lbs  j
IOT
3-10
POLY ENGINEERING
Trigonometry and Vectors
IOT
3-10
POLY ENGINEERING
Trigonometry and Vectors
IOT
3-10
POLY ENGINEERING
Trigonometry and Vectors
CLASSWORK / HOMEWORK
Complete problem #4 on the Vector Worksheet
IOT
3-10
POLY ENGINEERING