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Trigonometry and Vectors. REVIEW. Common triangles in Geometry and Trigonometry. You must memorize these triangles. 45 o. 60 o. 2. 1. 1. 30 o. 45 o. 1. 2. 3. Trigonometry and Vectors. REVIEW. Common triangles in Geometry and Trigonometry. 5. 3. 4. 1. Trigonometry and Vectors.

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Trigonometry and Vectors

REVIEW

Common triangles in Geometry and Trigonometry

You must memorize these triangles

45o

60o

2

1

1

30o

45o

1

2

3

Trigonometry and Vectors

REVIEW

Common triangles in Geometry and Trigonometry

5

3

4

1

Trigonometry and Vectors

opposite hypotenuse

sin A =

tan A =

cos A =

IOTPOLYENGINEERING

3-8

REVIEW

Trigonometric Functions

Trigonometric functions are ratios of the lengths of the segments that make up angles.

Trigonometry and Vectors

IOTPOLYENGINEERING

3-9

Vectors

• Scalar Quantities – a quantity that involves magnitude only; direction is not important
• Tiger Woods – 6’1”
• Shaquille O’Neill – 7’0”
• Vector Quantities – a quantity that involves both magnitudeand 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

Trigonometry and Vectors

IOTPOLYENGINEERING

3-9

Scalar or Vector?

• 400 mph due north
• \$100
• 10 lbs weight
• 5 miles northeast
• 6 yards
• 1000 lbs force

Magnitude and Direction

Magnitude and Direction

Vector

Vector

Magnitude only

Magnitude only

Scalar

Scalar

Magnitude and Direction

Magnitude only

Vector

Scalar

Trigonometry and Vectors

IOTPOLYENGINEERING

3-9

Vectors

• Free-body Diagram
• A diagram that shows all external forces acting on an object.

applied force

normal force

N

F

Ff

friction force

force of gravity (weight)

Wt

Trigonometry and Vectors

IOTPOLYENGINEERING

3-9

Vectors

• Describing vectors –
• We MUST represent both magnitudeand direction.
• Describe the force applied to the wagon by the skeleton:

Hat signifies vector quantity

40 lbs

45o

F = 40 lbs 45o

magnitude

direction

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Vectors – Scalar Multiplication

• We can multiply any vector by a real number.
• Original direction is maintained, new magnitude.

2

½

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

• 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)

March 14, 2011

IOTPOLYENGINEERING

3-10

Drill

Find a + b

Find 2 a

y

y

a

a

b

x

x

Drill

IOTPOLYENGINEERING

3-10

Find 2 a

y

2a

a

x

Drill

IOTPOLYENGINEERING

3-10

Find a + b

y

a+b

a

a

b

x

Drill

IOTPOLYENGINEERING

3-10

Find a + b

b

y

a

a+b

b

x

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Vectors – Rectangular Components

• 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 axis.

y

F

Fy

x

Fx

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Vectors – Rectangular Components

This means:

vector F = vector Fx + vector Fy

y

F

Fy

x

Fx

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Unit vector

Vectors – Rectangular Components

Vector Fx= Magnitude Fxtimes vector i

F = Fx i + Fyj

Fx= Fx i

i denotes vector in x direction

y

Vector Fy= Magnitude Fytimes vector j

F

Fy= Fy j

Fy

j denotes vector in y direction

x

Fx

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

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

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Vectors – Rectangular Components

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

y

F

Fy

x

Fx

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Vectors – Rectangular Components

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

cos q = Fx / F

Fx = F cos qi

sin q = Fy / F

Fy = F sin qj

F

Fy

q

Fx

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Vectors – Rectangular Components

When are Fx and Fy Positive/Negative?

Fy +

Fy +

y

F

Fx +

Fx -

F

x

F

F

Fx -

Fx +

Fy -

Fy -

IOTPOLYENGINEERING

3-10

Vectors – Rectangular Components

Complete the following chart in your notebook:

I

II

III IV

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Vectors – Rectangular Components

Each grid space represents 1 lb force.

What is Fx?

Fx = (-1 lbs)i

What is Fy?

Fy = (3 lbs)j

What is F?

F = (-1 lbs)i + (3 lbs)j

y

F

x

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Vectors – Resultant Forces

Resultant forces are the overall combination of all forces acting on a body. 1) Break up all forces into x and y component forces

2) add up all of the component forces in x-direction

3) add up all of the component forces in y-direction

4) Write resultant as single vector in rectangular components

150 lb

60o

100 lb

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

1) Break up all forces into x and y component forces

Space Junk:

150 lb

60o

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Break up all forces into x and y component forces

Gravity

100 lb

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

2) Add up all forces in x direction

150 lb

60o

100 lb

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

3) Add up all forces in y direction

150 lb

60o

100 lb

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

4) Write resultant as single vector in rectangular components

150 lb

60o

100 lb

Classwork

IOTPOLYENGINEERING

3-10

Complete Worksheet

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

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

= (75 lbs)i

SFx= (75 lbs)i

No x-component

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

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 lbs)j

Wy = -(100 lbs)j

SFy= (75 lbs)j - (100 lbs)j

SFy = (75 - 100 lbs)j

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

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 - 100 lbs)j

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

IOTPOLYENGINEERING

3-10

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Trigonometry and Vectors

IOTPOLYENGINEERING

3-10

Trigonometry and Vectors

CLASSWORK/ HOMEWORK

Complete problem #4 on the Vector Worksheet