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Vectors and Scalars

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A scalar is a quantity that is completely specified by a ... At point C, vx is negative. The slope is negative ... vx = v x avg = Dx / Dt. Also, ... – PowerPoint PPT presentation

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Title: Vectors and Scalars


1
Vectors and Scalars
  • A scalar is a quantity that is completely
    specified by a positive or negative number with
    an appropriate unit and has no direction.
  • A vector is a physical quantity that must be
    described by a magnitude (number) and appropriate
    units plus a direction.

2
Some Notes About Scalars
  • Some examples
  • Temperature
  • Volume
  • Mass
  • Time intervals
  • Rules of ordinary arithmetic are used to
    manipulate scalar quantities

3
Vector Example
  • A particle travels from A to B along the path
    shown by the dotted red line
  • This is the distance traveled and is a scalar
  • The displacement is the solid line from A to B
  • The displacement is independent of the path taken
    between the two points
  • Displacement is a vector

4
Other Examples of Vectors
  • Many other quantities are also vectors
  • Some of these include
  • Velocity
  • Acceleration
  • Force
  • Momentum

5
Vector Notation
  • When handwritten, use an arrow
  • When printed, will be in bold print with an
    arrow
  • When dealing with just the magnitude of a vector
    in print, an italic letter will be used A or
  • The magnitude of the vector has physical units
  • The magnitude of a vector is always a positive
    number

6
Equality of Two Vectors
  • Two vectors are equal if they have the same
    magnitude and the same direction
  • if A B and they point along parallel
    lines
  • All of the vectors shown are equal

7
Adding Vectors
  • When adding vectors, their directions must be
    taken into account
  • Units must be the same
  • Graphical Methods
  • Use scale drawings
  • Algebraic Methods
  • More convenient

8
Adding Vectors Graphically
  • Continue drawing the vectors tip-to-tail
  • The resultant is drawn from the origin of to
    the end of the last vector
  • Measure the length of and its angle
  • Use the scale factor to convert length to actual
    magnitude

9
Adding Vectors Graphically, final
  • When you have many vectors, just keep repeating
    the process until all are included
  • The resultant is still drawn from the origin of
    the first vector to the end of the last vector

10
Adding Vectors, Rules
  • When two vectors are added, the sum is
    independent of the order of the addition.
  • This is the commutative law of addition

11
Adding Vectors, Rules cont.
  • When adding three or more vectors, their sum is
    independent of the way in which the individual
    vectors are grouped
  • This is called the Associative Property of
    Addition

12
Adding Vectors, Rules final
  • When adding vectors, all of the vectors must have
    the same units
  • All of the vectors must be of the same type of
    quantity
  • For example, you cannot add a displacement to a
    velocity

13
Negative of a Vector
  • The negative of a vector is defined as the vector
    that, when added to the original vector, gives a
    resultant of zero
  • Represented as
  • The negative of the vector will have the same
    magnitude, but point in the opposite direction

14
Subtracting Vectors
  • Special case of vector addition
  • Continue with standard vector addition procedure

15
Multiplying or Dividing a Vector by a Scalar
  • The result of the multiplication or division is a
    vector
  • The magnitude of the vector is multiplied or
    divided by the scalar
  • If the scalar is positive, the direction of the
    result is the same as of the original vector
  • If the scalar is negative, the direction of the
    result is opposite that of the original vector

16
Multiplying Vectors
  • Two vectors can be multiplied in two different
    ways
  • One is the scalar product
  • Also called the dot product
  • The other is the vector product
  • Also called the cross product
  • These products will be discussed as they arise in
    the text

17
Components of a Vector
  • A component is a part
  • It is useful to use rectangular components
  • These are the projections of the vector along the
    x- and y-axes

18
Vector Component Terminology
  • are the component vectors of
  • They are vectors and follow all the rules for
    vectors
  • Ax and Ay are scalars, and will be referred to as
    the components of A
  • The combination of the component vectors is a
    valid substitution for the actual vector

19
Components of a Vector, 2
  • The x-component of a vector is the projection
    along the x-axis
  • The y-component of a vector is the projection
    along the y-axis
  • When using this form of the equations, q must be
    measured ccw from the positive x-axis

20
Components of a Vector, 3
  • The components are the legs of the right triangle
    whose hypotenuse is
  • May still have to find ? with respect to the
    positive x-axis
  • Use the signs of Ax and Ay

21
Components of a Vector, final
  • The components can be positive or negative and
    will have the same units as the original vector
  • The signs of the components will depend on the
    angle

22
Unit Vectors
  • A unit vector is a dimensionless vector with a
    magnitude of exactly 1.
  • Unit vectors are used to specify a direction and
    have no other physical significance

23
Unit Vectors, cont.
  • The symbols
  • represent unit vectors in the x, y and z
    directions
  • They form a set of mutually perpendicular vectors

24
Unit Vectors in Vector Notation
  • is the same as Ax and is the same as
    Ay etc.
  • The complete vector can be expressed as

25
Adding Vectors Using Unit Vectors
  • Using
  • Then
  • Then Rx Ax Bx and Ry Ay By

26
Adding Vectors with Unit Vectors Diagram
27
Adding Vectors Using Unit Vectors Three
Directions
  • Using
  • Rx Ax Bx , Ry Ay By and Rz Az Bz

  • etc.

28
Exercises of chapter 1
  • 1,4, 6, 8, 13, 17, 26, 35, 37, 43, 49, 52,
  • 57, 62, 64, 66, 68.

29
Chapter 2
  • Motion in One Dimension

30
Kinematics
  • To describe motion while ignoring the agents that
    caused the motion
  • For now, we will consider motion in one dimension
    - along a straight line -, and use the particle
    model.
  • A particle is a point-like object, has mass but
    infinitesimal size

31
Position
  • Defined in terms of a frame of reference
  • The reference frame must has an origin.
  • The objects position is its location with
    respect to the origin of the frame of reference.

32
Position-Time Graph
  • The position-time graph shows the motion of the
    particle (car)
  • The smooth curve is a guess as to what happened
    between the data points

33
Displacement
  • Defined as the change in position during some
    time interval Dt
  • Represented as ?x
  • Xi is the initial position and Xf is the final
    position.
  • Different than distance the length of a path
    followed by a particle

34
Average Velocity
  • The average velocity is rate at which the
    displacement occurs
  • The dimensions are length / time L/T
  • The slope of the line connecting the initial and
    final points in the position time graph

35
Average Velocity, cont
  • Gives no details about the motion
  • Gives the result of the motion
  • It can be positive or negative
  • It depends on the sign of the displacement
  • It can be interpreted graphically

36
Displacement from a Velocity - Time Graph
  • The displacement of a particle during the time
    interval ti to tf is equal to the area under the
    curve between the initial and final points on a
    velocity-time curve

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Instantaneous Velocity
  • The limit of the average velocity as the time
    interval becomes infinitesimally short, or as the
    time interval approaches zero
  • The instantaneous velocity indicates what is
    happening at every point of time

44
Instantaneous Velocity, graph
  • The instantaneous velocity is the slope of the
    line tangent to the x vs. t curve
  • This would be the green line
  • The blue lines show that as ?t gets smaller, they
    approach the green line

45
Instantaneous Velocity, equations
  • The general equation for instantaneous velocity
    is
  • When velocity is used in the text, it will mean
    the instantaneous velocity

46
Instantaneous Velocity, Signs
  • At point A, vx is positive
  • The slope is positive
  • At point B, vx is zero
  • The slope is zero
  • At point C, vx is negative
  • The slope is negative

47
Instantaneous Speed
  • The instantaneous speed is the magnitude of the
    instantaneous velocity vector
  • Speed can never be negative

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Fig 2.5
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Constant Velocity
  • If the velocity of a particle is constant
  • Its instantaneous velocity at any instant is the
    same as the average velocity over a given time
    period
  • vx v x avg Dx / Dt
  • Also, xf xi vx t
  • These equations can be applied to particles or
    objects that can be modeled as a particle moving
    under constant velocity

59
Fig 2.6
60
Average Acceleration
  • Acceleration is the rate of change of the
    velocity
  • It is a measure of how rapidly the velocity is
    changing
  • Dimensions are L/T2
  • SI units are m/s2
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