Physics and Physical Measurement

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Physics and Physical Measurement

• Topic 1.3 Scalars and Vectors

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Scalars Quantities

• Scalars can be completely described by magnitude
  (size)
• Scalars can be added algebraically
• They are expressed as positive or negative
  numbers and a unit
• examples include - mass, electric charge,
  distance, speed, energy

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Vector Quantities

• Vectors need both a magnitude and a direction to
  describe them (also a point of application)
• When expressing vectors as a symbol, you need to
  adopt a recognized notation
• e.g.
• They need to be added, subtracted and multiplied
  in a special way
• Examples - velocity, weight, acceleration,
  displacement, momentum, force

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Addition and Subtraction

• The Resultant (Net) is the result vector that
  comes from adding or subtracting a number of
  vectors
• If vectors have the same or opposite directions
  the addition can be done simply
• same direction add
• opposite direction subtract

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Co-planar vectors

• The addition of co-planar vectors that do not
  have the same or opposite direction can be solved
  by using scale drawings to get an accurate
  resultant
• Or if an estimation is required, they can be
  drawn roughly
• or by Pythagoras theorem and trigonometry
• Vectors can be represented by a straight line
  segment with an arrow at the end

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Triangle of Vectors

• Two vectors are added by drawing to scale and
  with the correct direction the two vectors with
  the tail of one at the tip of the other.
• The resultant vector is the third side of the
  triangle and the arrow head points in the
  direction from the free tail to the free tip

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Example
R a b
a
b


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Parallelogram of Vectors

• Place the two vectors tail to tail, to scale and
  with the correct directions
• Then complete the parallelogram
• The diagonal starting where the two tails meet
  and finishing where the two arrows meet becomes
  the resultant vector

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Example
R a b
a
b


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More than 2

• If there are more than 2 co-planar vectors to be
  added, place them all head to tail to form
  polygon when the resultant is drawn from the
  free tail to the free tip.
• Notice that the order doesnt matter!

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Subtraction of Vectors

• To subtract a vector, you reverse the direction
  of that vector to get the negative of it
• Then you simply add that vector

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Example
a
b
-

R a (- b)
-b
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Multiplying Scalars

• Scalars are multiplied and divided in the normal
  algebraic manner
• Do not forget units!
• 5m / 2s 2.5 ms-1
• 2kW x 3h 6 kWh (kilowatt-hours)

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Multiplying Vectors

• A vector multiplied by a scalar gives a vector
  with the same direction as the vector and
  magnitude equal to the product of the scalar and
  a vector magnitude
• A vector divided by a scalar gives a vector with
  same direction as the vector and magnitude equal
  to the vector magnitude divided by the scalar
• You dont need to be able to multiply a vector by
  another vector

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Resolving Vectors

• The process of finding the Components of vectors
  is called Resolving vectors
• Just as 2 vectors can be added to give a
  resultant, a single vector can be split into 2
  components or parts

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The Rule

• A vector can be split into two perpendicular
  components
• These could be the vertical and horizontal
  components

Vertical component
Horizontal component
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• Or parallel to and perpendicular to an inclined
  plane

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• These vertical and horizontal components could be
  the vertical and horizontal components of
  velocity for projectile motion
• Or the forces perpendicular to and along an
  inclined plane

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Doing the Trigonometry
V
Sin ? opp/hyp y/?V?
y
Therefore y ?V?sin ? In this case this is the
vertical component
?
x
Cos ? adj/hyp x/?V?
Therefore x ?V?cos ? In this case this is the
horizontal component
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Quick Way

• If you resolve through the angle it is
• cos
• If you resolve not through the angle it is
• sin

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Adding 2 or More Vectors by Components

• First resolve into components (making sure that
  all are in the same 2 directions)
• Then add the components in each of the 2
  directions
• Recombine them into a resultant vector
• This will involve using Pythagoras theorem

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Question

• Three strings are attached to a small metal ring.
  2 of the strings make an angle of 70o and each is
  pulled with a force of 7N.
• What force must be applied to the 3rd string to
  keep the ring stationary?

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Answer

• Draw a diagram

7 cos 35o 7 cos 35o
70o
7 sin 35o
7 sin 35o
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• Horizontally
• 7 sin 35o - 7 sin 35o 0
• Vertically
• 7 cos 35o 7 cos 35o F
• F 11.5N
• And at what angle?
• 145o to one of the strings.