CHAPTER 5 FORCES IN TWO DIMENSIONS

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CHAPTER 5 FORCES IN TWO DIMENSIONS

•  
• In this chapter you will
•  
•   Represent vector quantities both graphically
  and algebraically.
• Use Newtons Laws to analyze motion when Friction
  is involved.
• Use Newtons Laws and your knowledge of vectors
  to analyze motion in 2 dimensions.

2
CHAPTER 5 SECTIONS

• Section 5.1 Vectors
• Section 5.2 Friction
• Section 5.3 Force and Motion in Two Dimensions

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SECTION 5.1 VECTORS

• Objectives
• Evaluate the sum of 2 or more vectors in 2
  dimensions graphically.
• Determine the components of vectors.
• Solve for the sum of 2 or more vectors
  algebraically by adding the components of vectors.

4
VECTORS REVISITED

• A vector quantity can be represented by an arrow
  tipped line segment. -----gt
• Resultant vector sum of 2 or more vectors.

5
VECTORS IN MULTIPLE DIMENSIONS

• The vectors are added by placing the tail of one
  vector at the head of the other vector.
•  
• The resultant is drawn from the tail of the first
  vector to the head of the last vector.
•  
• To find the magnitude of the resultant measure
  its length using the same scale used to draw the
  2 vectors.
• Its direction can be found with a protractor.
  The direction is expressed as an angle measured
  counterclockwise from the horizontal.

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VECTORS IN MULTIPLE DIMENSIONS

• In the example in Figure 6-2 we have 95 m east
  and 55 m north. Our resultant would be 110 m at
  30? north of east. In this problem since we get
  a right triangle we can use a2 b2 c2.
• Force vectors are added in the same way as
  position or velocity vectors.
• 90?
• II I
• 180 ----------------------------0? , 360?
• III IV
• 270?

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VECTORS IN MULTIPLE DIMENSIONS

• The resultant is drawn from the tail of the first
  vector to the head of the last vector.
• The angle is found with a protractor.
• You can Add vectors by placing them Tip-to-Tail
  (or Head to Tail) and then drawing the Resultant
  of the vector by connecting the TAIL OF THE FIRST
  VECTOR TO THE TIP OF THE LAST VECTOR.

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VECTORS IN MULTIPLE DIMENSIONS

• Go Over Figure 5.2 p. 120
• Because the length and direction are the only
  important characteristics of the vector, the
  vector is unchanged by this movement. This is
  always true for this type of movement.
• To find the Resultant you measure it to get its
  Magnitude and use a Protractor to get its
  direction.
• Pythagorean Theorem if you have a right
  triangle then you can find the lengths of the
  sides using a2 b2 c2 and here we can use A2
  B2 R2

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VECTORS IN MULTIPLE DIMENSIONS

• Law of Cosines - The square of the magnitude of
  the resultant vector is equal to the sum of the
  magnitude of the squares of the two vectors,
  minus two times the product of the magnitudes of
  the vectors, multiplied by the cosine of the
  angle between them.
• R2 A2 B2 2AB cos?
• Law of Sines - The magnitude of the resultant,
  divided by the sine of the angle between two
  vectors, is equal to the magnitude of one of the
  vectors divided by the angle between that
  component vector and the resultant vector.
• __R__ __A__ __B__
• sin ? sin a sin b

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VECTORS IN MULTIPLE DIMENSIONS

• Do Example Problem 1 p. 121
• A) A2 B2 R2 B) R2 A2 B2 2AB cos?
• 152 252 R2 R2 152 252
  2(15)(25)cos(135)
• 225 625 R2 R2 225 625 750(-.707)
• 850 R2 R2 850 530.33
• 29.155 km R R2 1380.33
• R 37.153 km
•  
• OR Use Components of Vectors

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VECTORS IN MULTIPLE DIMENSIONS

• B) OR Use Components of Vectors 
• A1x A cos ? A1y A sin ? Then
• A1x 15 cos(0) A1y 15 sin(0) A2 B2
  R2
• A1x 15 (1) A1y 15 (0)
  (32.68)2 17.682 R2
• A1x 15 A1y 0 1067.9824
  312.5824 R2
• A2x A cos ? A2y A sin ?
  1380.5648 R2
• A2x 25 cos(45) A2y 25 sin(45)
  37.156 km R
• A2x 25 (.707) A2y 25 (.707)
• A2x 17.68 A2y 17.68
• Ax A1x A2x Ay A1y A2y
• Ax 15 17.68 Ay 0 17.68
• Ax 32.68 km Ay 17.68 km
• Do Practice Problems p. 121 1-4

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COMPONENTS OF VECTORS

• Trigonometry branch of math that deals with the
  relationships among angles and sides of
  triangles.
•  
• Sine (sin) opposite side over the hypotenuse
  sin ? opp / hyp
•  
• Cosine (cos) adjacent side over the hypotenuse
  cos ? adj /hyp
•  
• Tangent (tan) opposite side over the adjacent
  side tan ? opp/adj
• SOH CAH TOA (sin opp/hyp cos adj/hyp
  tan opp/adj

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COMPONENTS OF VECTORS

• We have seen that 2 or more vectors acting in
  different directions from the same point may be
  replaced by a single vector, the RESULTANT. The
  resultant has the same effect as the original
  vectors
•  
• Components of the Vector the 2 perpendicular
  vectors that can be used to represent a single
  vector. You break a vector down into its
  horizontal and vertical parts.
• A Ax Ay

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COMPONENTS OF VECTORS

• Vector Resolution the process of breaking a
  vector into its components. Note the original
  vector is the hypotenuse of a right triangle and
  thus larger than both components.
• Fh F cos ? or Fx F cos ?
• Fv F sin ? or Fy F sin ?
• Or
• Ax A cos ?
• Ay A sin ?

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COMPONENTS OF VECTORS

• Go Over figure 5.4 p. 122
•   90?
• Ax lt 0 Ax gt 0
• Ay gt 0 Ay gt 0
• II I
• 180 -----------------------------0? , 360?
• III IV
• Ax gt 0 Ax gt 0
• Ay lt 0 Ay lt 0
• 270?

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COMPONENTS OF VECTORS

• If the value for Ay is positive then it moves up
  and if Ax is positive then it moves to the right.
  If Ay is negative it moves down and if Ax is
  negative it moves left.

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ALGEBRAIC ADDITION OF VECTORS

• Angles do not have to be perpendicular in order
  to use vector resolution.
•  
• What you do first is make each vector into its
  perpendicular components (Components of the
  Vector).
•  
• Then the vertical components are added together
  to produce a single vector that acts in the
  vertical direction.
• Next all of the horizontal components are added
  together to produce a single vector that acts in
  the horizontal direction.

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ALGEBRAIC ADDITION OF VECTORS

• The resulting vertical and horizontal components
  can be added together to obtain the Final
  Resultant. By using Pythagorean Theorem where
  the Horizontal and Vertical Components are youre
  a and b and you find R.
•  
• Angle of the Resultant Vector equals the
  inverse tangent of the quotient of the y
  component divided by the x component of the
  resultant vector.
• tan ? Ay / Ax or ? tan-1(Ay / Ax)

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ALGEBRAIC ADDITION OF VECTORS

• Note that when tan ? gt 0, most calculators give
  the angle between 0 and 90, and when tan ? lt 0,
  the angle is reported to be between 0 and -90.
• PROBLEM SOLVING STRATEGIES p. 123
• Choose a coordinate system (0 always due East)
• Resolve the vectors into their x-components using
  Ax A cos ? and their y-components using Ay A
  sin ? where ? is the angle measured
  Counterclockwise from the positive x-axis.
• Add or subtract the component vectors in the
  x-direction.
• Add or subtract the component vectors in the
  y-direction.
• Use the Pythagorean Theorem to find the Magnitude
  of the resultant vector.
• To find the Angle of the resultant vector use tan
  ? Ay / Ax or ? tan-1(Ay / Ax)

20
ALGEBRAIC ADDITION OF VECTORS

• Do Example 2 p. 124
• A1x A cos ? A1y A sin ? Then
• A1x 5 cos(90) A1y 5 sin(90)
  A2 B2 R2
• A1x 5 (0) A1y 5 (1)
  (-11.49)2 4.642 R2
• A1x 0 A1y 5
  132.0201 21.5296 R2
• Ax A cos ? Ay A sin ?
  153.5497 R2
• Ax 15 cos(140) Ay 15 sin(140)
  12.39 km R
• Ax 15 (-.766) Ay 15 (.643)
• Ax -11.49 Ay 9.64
  tan ? Ay / Ax
• tan ? 4.64 / -11.49
• Ax A1x A2x Ay A1y A2y
  tan ? -.4038
• -11.49 0 A2x 9.64 5 A2y
  INV TAN
• -11.49 A2x 4.64 A2y
  ? -21.99
• or 180 21.99
  158.01

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ALGEBRAIC ADDITION OF VECTORS

• Do Practice Problems p. 125 5-10
•  
• Do 5.1 Section Review p. 125 11-16