Vectors Ch 6

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Vectors Ch 6
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Vectors

• Vectors are arrows
• They have both size and direction
• Vectors have NO PLACE!!!

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Scalars

• Scalars are line segments (no direction)

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Vector Examples Velocity
24 m/s
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Force
980 N
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Combining Vectors
If two or more vectors are acting on an object,
it is possible to combine (add) them as long as
they are drawn head- to -tail
head-
tail
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Vectors in the Same Direction
When vectors are in the same direction, their
magnitudes combine and the overall value
increases.
For example
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7 m/s
10 m/s


Resultant vector
17 m/s
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Vectors in Opposite Directions
When vectors are in the opposite directions,
their magnitudes combine and the overall value
decreases. The resultant vector is in the
direction of the larger vector
10
16 m/s
10 m/s


6 m/s
Resultant vector
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More than two

• It is possible to combine more than two vectors
• Think of a tug of war..

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If all of these people are pulling,
Then their strengths will combine
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7 N
6 N
4 N
12 N
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For a total strength of
29 N
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It is possible to combine vectors that are moving
in directions at angles to each other.
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For example, what would be the effect of these
two vectors acting simultaneously? (still draw
head-tail)
9 N
12 N
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?
9 N
12 N
From the Pythagorean theorem c2a2b2 the length
of the resultant is sq rt(12292)15 From trig
the tan-1(9/12) 37 degrees So the complete
answer is 15 N 37 degrees from the 12 N vector
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• Remember to draw head-to-tail !!!

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Either way is ok
12 N
15 N
9 N
9 N
12 N
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Right angles are always easier to work with
With the proper technique any combination of
vectors can be resolved
21 m
12 N
12 N
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Vector Terminology

• 1) Components the two vectors (usually a
  horizontal and a vertical) that can be used to
  replace a larger vector.
• resolution the process of finding the
  two components of a vector.
• 2) Resultant the sum of at least two other
  vectors.
• 3) Equilibrium the condition established when
  the resultant of the forces is zero and Newtons
  1st Law is observed

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Basic Trigonometry Functions

• sin q opposite/hypotenuse soh
• cos q adjacent/hypotenuse cah
• tan q opposite/adjacent toa

sin q opposite/hypotenuse soh cos q
adjacent/hypotenuse cah tan q
opposite/adjacent toa
hypotenuse
opposite
q
adjacent
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Example using Pythagorean Theorem and Tan function

• An archaeologist climbs the Great Pyramid in
  Giza, Egypt. If the pyramids height is 136m and
  its width is 2.3x102m, what is the magnitude and
  the direction of the archaeologists displacement
  while climbing from the bottom of the pyramid to
  the top?

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• YOU TRY PG. 91 1-4

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Example using Sine and Cosine functions

• Lets examine a scene from a new action movie.
  For this scene a biplane travels at 95 km/hr at
  an angle of 20 degrees relative to the ground.
  Attempting to film the plane from below, a camera
  team travels in a truck, keeping the truck
  beneath the plane at all times. How fast must
  the truck travel to remain directly below the
  plane?

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• YOU TRY PG. 94 1-7

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Solving Vector Problems

• 1) Graphically (parallelogram method) a ruler
  and protractor are used to draw all vectors to
  scale from head-tail. The answer is the
  resultant which is drawn from the tail of the
  first vector to the head of the last vector.

r
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• 2) Trigonometrically
• a) using trig. functions resolve each vector
  into its horizontal and vertical components.
• b) draw a super-sized right triangle from all
  of the combined horizontal and vertical
  components.
• c) use the Pythagorean theorem to determine the
  resultant (hypotenuse) AND trig. functions to
  determine the angle of the resultant.

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The result is the same!!
r
c2a2b2
r
vertical
horizontal
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Example of adding vectors that are not
perpendicular

• A hiker walks 25.5 km from her base camp at 35
  degrees south of east. On the second day, she
  walks 41.0 km in a direction 65 degrees north of
  east, at which point she discovers a forest
  rangers tower. Determine the magnitude and
  direction of her resultant displacement between
  the base camp and the rangers tower.

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• YOU TRY PG. 97 1-4