Chapter 3 Two-Dimensional Motion and Vectors

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Chapter 3 Two-Dimensional Motion and Vectors
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3-1 Introduction to Vectors

• Vectors indicate direction scalars do not.

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Scalar

• A physical quantity that has only magnitude, but
  not direction.
• Ex speed 30 km/hr

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Vector

• A physical quantity that has both magnitude and
  direction.
• Ex velocity 12 km/hr, North

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Resultant

• A vector representing the sum of two or more
  vectors.
• It starts from where the 1st one begins and the
  last one ends.

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The book uses symbols to distinguish between them.
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• Vectors by boldface type
• v 3.5 m/s, north
• Scalars by italicized type
• v 3.5 m/s

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Vectors are represented as arrows pointing in the
direction of the vector.

• The length is a representation of the vectors
  magnitude.

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When drawing vectors you need to establish a
scale.

• Ex 4500 m, west.
• You might chose
• 1 cm 500 m ? 9 cm
• or 1 in 1500 m ? 3 in

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We are going to add vectors by placing them head
to tail.
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Adding vectors along the same line

• If two or more forces are acting in the same
  direction, we simply add them.

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Example A force of 30 N is acting toward the
right and a force of 40 N is also applied to the
right, what is the Net Force?
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30 N 40 N 70 N

• To draw a representation of the of the forces or
  a vector diagram, we need to establish a scale.
• Scale 1 inch 10 N

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So the the first vector would be 3 inches.

• And the second vector would be 4 inches.

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So the the first vector would be 3 inches.

• And the second vector would be 4 inches.

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So the the first vector would be 3 inches.

• And the second vector would be 4 inches.

70 N
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When vectors are added in opposite directions we
subtract them.

• Example A 40 N force acting east and a 60 N
  force acting west.

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Establish a scale and place head to tail.

• Pick a starting point and draw vector to scale.
• Scale 1 inch 10 N

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Establish a scale and place head to tail.

• Pick a starting point and draw vector to scale.
• Scale 1 inch 10 N

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Establish a scale and place head to tail.

• Pick a starting point and draw vector to scale.
• Scale 1 inch 10 N

20 N west
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Vectors acting at right angles

• 1. Pick a scale.
• 2. Draw first vector to scale.
• 3. Draw second vector, starting at the head of
  the first one.

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• The resultant is the arrow that is created by
  drawing a line from the tail of the first vector
  to the head of the second vector.

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Example A planes engine applies a force of 200
N, north and a wind blows the plane with a force
of 40 N east. What is the net force?
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Scale 2 inches 40 N
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Scale 2 inches 40 N
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Scale 2 inches 40 N

• We can measure the length of the gray arrow in
  inches and convert it to Newtons, using the
  conversion scale.

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Another way to calculate the net force or
resultant is to use Pythagorean theorem.

• c2 a2 b2
• c
• b
• a

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c2 a2 b2

• c2 2002 402
• c2 40000 N 1600 N
• c2 41600 N
• c square root of 41600 N
• c 203.96 N
• Hypotenuse of a triangle is always the longest
  side.

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Ex 1 A, B, C Add the following vectors.

• 240 m/s, West 600 m/s, West
• B) 300 m, East 120 m, West
• C) 360 m/s, South 270 m/s, West

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Right Triangles
B
c
a
A
b
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1. Side c is called the hypotenuse.
2. Sides a and b called legs.
3. The leg across from an angle is called the
   opposite side.
4. The leg next to an angle is called the adjacent
   side.

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Pythagorean Theorem
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Q Greek letter Theta

• This is the symbol used in Trig and Geometry to
  represent an angle.
• Q 23O

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B
c
a
A
b
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B
c
a
A
b
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B
c
a
A
b
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When given a vector at any angle, use the trig
functions to find the x and y components.
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Components of Vectors
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Ex 4 Find the x and y components for 72 m/s _at_
30O (pick scale)
72 m/s
30o
Scale 1 inch 12 m/s
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Ex 2 Find the x and y components for 72 m/s _at_
30O (pick scale)
72 m/s
y
30o
x
Scale 1 inch 12 m/s
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x-component

• Use the cosine function, because you have the
  angle and the hypotenuse and its the side
  adjacent.

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x-component

• Use the cosine function, because you have the
  angle and the hypotenuse and its the side
  adjacent.

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x-component

• Use the cosine function, because you have the
  angle and the hypotenuse and its the side
  adjacent.

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y-component

• Use the sine function, because you have the angle
  and the hypotenuse and its the side opposite.

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y-component

• Use the sine function, because you have the angle
  and the hypotenuse and its the side opposite.

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y-component

• Use the sine function, because you have the angle
  and the hypotenuse and its the side opposite.

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Or since we knew the hypotenuse and x-component,
we could have used the Pythagorean Theorem.
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Ex 5 140 m _at_ 145o. Find the x and y components.
145o
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145o
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Still use the given angle, even though the
triangle formed doesnt have this angle. This
will ensure the proper sign for the x and y
components.
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x-component
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x-component
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x-component
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y-component
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y-component
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y-component
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When adding 2 or more vectors at different angles.

1. Pick a scale for both vectors, draw the1st
   vector.
2. Draw the 2nd vector to scale, starting where the
   1st one ended.

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• 3. Find the x and y components for each vector.
• 4. Draw the final vector, from where the 1st
  vector started to where the 2nd one ends.

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• 5. Add the x components together, then the y
  components.
• 6. Use the added x and y components to find
  the resultant (Pythagorean Theorem) and its angle
  (Inverse tangent).

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v1
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v1
v2
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v1
v2
q2
v1,y
v1,x
q1
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v1
v2
v2,y
q2
v2,x
v1,y
v1,x
q1
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v1
v2
v2,y
vR
q2
v2,x
v1,y
v1,x
q1
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v2,y
vR
v1,y
v1,x
v2,x
qR
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vRx v1,x v2,x
vRy v1,y v2,y
vR
qR
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Ex 4 Find the resultant vector and its angle for
4500 N at 70o 7500 N at 40o.

• Scale 1 cm 500 N

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• v1,x hyp cosQ
• v1,x 4500cos70
• v1,x 1539.1 N

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• v1,y hyp sinQ
• v1,y 4500sin70
• v1,y 4228.6 N

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• v2,x hyp cosQ
• v2,x 7500cos40
• v2,x 5745.3N

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• v2,y hyp sinQ
• v2,y 7500sin40
• v2,y 4820.9 N

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• vR,x v1,x v2,x
• vR,x 1539.1 N 5745.3 N
• vR,x 7284.4 N

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• vR,y v1,y v2,y
• vR,y 4228.6 N 4820.9 N
• vR,y 9049.5 N

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Ex 5 Find the resultant and its angle for 4 m/s
_at_ 47o east of south 8 m/s _at_ 15o south of west.
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v1
Q1
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v1
Q2
v2
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QR
v1
vR
v2
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• v1,x hyp sinQ
• v1,x 4sin47
• v1,x 2.93 m/s

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• v1,y hyp cosQ
• v1,y 4cos47
• v1,y -2.73 m/s

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• v2,x hyp cosQ
• v2,x 8cos15
• v2,x -7.73 m/s

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• v2,y hyp sinQ
• v2,y 8sin15
• v2,y -2.07 m/s

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• vR,x v1,x v2,x
• vR,x 2.93 m/s -7.73 m/s
• vR,x -4.80 m/s

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• vR,y v1,y v2,y
• vR,y -2.73 m/s -2.07 m/s
• vR,y -4.80 m/s

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