Vector_mechanics [Compatibility Mode]

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    Vectors

    Vector classifications:

    - Fixedorboundvectors have well defined points of

    application that cannot be changed without affecting

    an analysis.

    - Free vectors may be freely moved in space without

    changing their effect on an analysis.

    Equal vectors have the same magnitude and direction.

    Negative vector of a given vector has the same magnitude

    and the opposite direction.

    Engineers Mechanics- Review of Vector Algebra/Applications

    Addition of Vectors

    Parallelogram rule for vector addition

    Triangle rule for vector addition

    B

    B

    C

    C

    QPR

    BPQQPR

    cos2222 Law of cosines,

    Law of sines,

    P

    C

    R

    B

    Q

    A sinsinsin

    Vector addition is commutative,

    PQQP

    Vector subtraction

    Engineers Mechanics- Review of Vector Algebra/Applications

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    Addition of Vectors

    Addition of three or more vectors through

    repeated application of the triangle rule

    The polygon rule for the addition of three or

    more vectors.

    Vector addition is associative,

    SQPSQPSQP

    Multiplication of a vector by a scalar

    Engineers Mechanics- Review of Vector Algebra/Applications

    Engineers Mechanics- Review of Vector Algebra/Applications

    Addition of Vectors

    A quantity which has magnitude and

    direction, but doesnt follow

    parallelogram law, cannot be a vector.

    Can you name such a quantity?

    Think in terms of associative property!!

    Answer: Finite Rotation

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    Application: Resultant of Several Concurrent Forces Concurrent forces: set of forces which all

    pass through the same point.

    A set of concurrent forces applied to a

    particle may be replaced by a single

    resultant force which is the vector sum of the

    applied forces.

    Vector force components: two or more force

    vectors which, together, have the same effect

    as a single force vector.

    Engineers Mechanics- Review of Vector Algebra/Applications

    Rectangular Components of a Vector

    cosFFx

    Vector components may be expressed as products of

    the unit vectors with the scalar magnitudes of the

    vector components.

    Fx andFy are referred to as the scalar components of

    jFiFF yx

    F

    May resolve a vector into perpendicular components so

    that the resulting parallelogram is a rectangle.

    are referred to as rectangular vector components and

    yx FFF

    yx FF

    and

    Define perpendicularunit vectors which are

    parallel to thex andy axes.ji

    and

    sinFFy

    Engineers Mechanics- Review of Vector Algebra/Applications

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    Application: Addition of Concurrent Forces

    SQPR

    Wish to find the resultant of 3 or more

    concurrent forces,

    jSQPiSQPjSiSjQiQjPiPjRiR

    yyyxxx

    yxyxyxyx

    Resolve each force into rectangular components

    x

    xxxx

    F

    SQPR

    The scalar components of the resultant are

    equal to the sum of the corresponding scalar

    components of the given forces.

    y

    yyyy

    F

    SQPR

    x

    yyx

    R

    RRRR

    122 tan

    To find the resultant magnitude and direction,

    Engineers Mechanics- Review of Vector Algebra/Applications

    Example-1

    Two structural members A and B are bolted to

    a bracket as shown. Knowing that both

    members are in compression and that the force

    is 20 kN in member A and 30 kN in member B,

    determine, using trigonometry, the magnitude

    and direction of the resultant of the forces

    applied to the bracket by members A and B.

    SOLUTION KEY

    o Construct the force triangle

    and apply the sine and

    cosine rules.

    Engineers Mechanics- Review of Vector Algebra/Applications

    Example of a timber truss joint

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    SOLUTION

    Engineers Mechanics- Review of Vector Algebra/Applications

    Example-1

    A collar that can slide on a vertical rod is subjected to

    the three forces shown. Determine (a) the value of the

    angle for which the resultant of the three forces is

    horizontal, (b) the corresponding magnitude of the

    resultant.

    SOLUTION KEY

    o Since the resultant (R) is to be

    horizontal, sum of the vertical

    comp. of the forces, i.e., Ry = 0.Example of an Umbrella

    Engineers Mechanics- Review of Vector Algebra/Applications

    Example-2

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    1 - 11

    0y yR F

    90 lb 70 lb sin 130 lb cos 0

    13 cos 7 sin 9

    213 1 sin 7 sin 9

    2 2169 1 sin 49 sin 126 sin 81 2218 sin 126 sin 88 0

    sin 0.40899

    1.24

    (a) Since R is to be horizontal,Ry = 0

    Then,

    Squaringboth sides:

    Solving by quadratic formula:

    or,

    SOLUTION

    Engineers Mechanics- Review of Vector Algebra/Applications

    Example-2

    1 -12

    117.0 lbR or,

    Engineers Mechanics- Review of Vector Algebra/Applications

    Example-2

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    Rectangular Components of a Vector in Space

    1coscoscos222 zyx

    Components of the vectorF

    kji

    FkjiF

    kFjFiFF

    FFFFFF

    zyx

    zyx

    zyx

    zzyyxx

    coscoscos

    coscoscos

    coscoscos

    is a unit vector along the line of action of

    and are the direction

    cosines for

    F

    F

    zyx cosand,cos,cos222

    zyx FFFF

    Engineers Mechanics- Review of Vector Algebra/Applications

    Application: Rectangular Components of a Force Vector in Space

    222

    zyx dddd

    The magnitude of the force vector is

    F and the direction of the force is

    defined by the location of two

    points,

    222111 ,,and,, zyxNzyxM

    d

    FdF

    d

    FdF

    d

    FdF

    kdjdidd

    FF

    zzdyydxxdkdjdid

    NMd

    zz

    yy

    xx

    zyx

    zyx

    zyx

    1

    andjoiningvector

    121212

    d

    d

    d

    d

    d

    d zz

    y

    yx

    x cos;cos;cos

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    Scalar Product of Two Vectors The scalar productordot productbetween

    two vectors P andQ is defined as

    resultscalarcosPQQP

    Scalar products:

    - are commutative,

    - are distributive,

    - are not associative,

    PQQP

    2121 QPQPQQP

    undefined SQP

    Scalar products with Cartesian unit components,

    000111 ikkjjikkjjii

    kQjQiQkPjPiPQP zyxzyx

    2222 PPPPPP

    QPQPQPQP

    zyx

    zzyyxx

    Engineers Mechanics- Review of Vector Algebra/Applications

    Applications: Scalar Product of Two Force Vectors

    Angle between two force vectors:

    PQ

    QPQPQP

    QPQPQPPQQP

    zzyyxx

    zzyyxx

    cos

    cos

    Projection of a force vector on a given axis:

    OL

    OL

    PPQ

    QP

    PQQP

    OLPPP

    cos

    cos

    alongofprojectioncos

    zzyyxx

    OL

    PPP

    PP

    coscoscos

    - For an axis defined by a unit vector ():

    Q

    Q

    Note:

    Engineers Mechanics- Review of Vector Algebra/Applications

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    Vector Product of Two Vectors

    Vector product of two vectorsP andQ is defined

    as the vectorVwhich satisfies the following

    conditions:

    1. Line of action ofVis perpendicular to plane

    containingP andQ.

    2. Magnitude ofVis

    3. Direction ofVis obtained from the right-hand

    rule.

    sinQPV

    Vector products:

    - are not commutative,

    - are distributive,

    - are not associative,

    QPPQ

    2121 QPQPQQP

    SQPSQP

    Engineers Mechanics- Review of Vector Algebra/Applications

    Vector Products: Rectangular Components

    Vector products of Cartesian unit vectors,

    0

    0

    0

    kkikjjki

    ijkjjkji

    jikkijii

    Vector products in terms of rectangular

    coordinates

    kQjQiQkPjPiPV zyxzyx

    kQPQP

    jQPQPiQPQP

    xyyx

    zxxzyzzy

    zyx

    zyx

    QQQ

    PPP

    kji

    Engineers Mechanics- Review of Vector Algebra/Applications

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    Application: Moment of a Force About a Point Moment of a force produces a turning action on a

    rigid body

    The momentof a force F about O is defined as

    FrMO

    Engineers Mechanics- Review of Vector Algebra/Applications

    r is the position vector of A from O

    The moment vectorMO is perpendicular to the

    plane containing O and the forceF.

    Magnitude ofMO measures the tendency of the forceto cause rotation of the body about an axis alongMO.

    dis the perpendicular distance of the line of action

    of Force F from O. The sense of the moment may be

    determined by the right-ha