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  • 8/7/2019 Dynamics_Lecture8 [Compatibility Mode]

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    Plane Motion of Rigid Bodies:

    Energy and Momentum

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    Introduction

    Method of work and energy and the method of impulse and

    momentum will be used to analyze the plane motion of rigid

    bodies and systems of rigid bodies.

    Principle of work and energy is well suited to the solution of

    problems involving displacements and velocities.

    2211 TUT =+

    Principle of impulse and momentum is appropriate for

    problems involving velocities and time.

    ( ) ( )2121

    2

    1

    2

    1

    O

    t

    t

    OO

    t

    t

    HdtMHLdtFL

    =+=+

    Problems involving eccentric impact are solved by supplementing

    the principle of impulse and momentum with the application of

    the coefficient of restitution.

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    Principle of Work and Energy for a Rigid Body

    Method of work and energy is well adapted to

    problems involving velocities and displacements.

    Main advantage is that the work and kinetic energy

    are scalar quantities.

    Assume that the rigid body is made of a large

    number of particles.

    2211 TUT =+

    21,

    =21U

    particles forming body

    total work of internal and external forces

    acting on particles of body.

    Internal forces between particlesA andB are equal

    and opposite.

    Therefore, the net work of internal forces is zero.

    In general, small displacements of the particlesA

    andB are not equal but the components of the

    displacements alongAB are equal.

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    Work of Forces Acting on a Rigid Body

    Work of a force during a displacement of its

    point of application,

    ( ) ==2

    1

    2

    1

    cos21

    s

    s

    A

    A

    dsFrdFU

    Consider the net work of two forces

    forming a couple of moment during a

    displacement of their points of application.

    FF

    and

    M

    dM

    dFrdsF

    rdFrdFrdFdU

    =

    ==

    +=

    2

    211

    ( )12

    212

    1

    =

    =

    M

    dMU

    ifMis constant.

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    Work of Forces Acting on a Rigid Body

    Forces acting on rigid bodies which do no work:

    Forces applied to fixed points:

    - reactions at a frictionless pin when the supported body

    rotates about the pin.

    Forces acting in a direction perpendicular to the displacement

    of their point of application:

    - reaction at a frictionless surface to a body moving alongthe surface

    - weight of a body when its center of gravity moves

    horizontally

    Friction force at the point of contact of a body rolling withoutsliding on a fixed surface.

    ( ) 0=== dtvFdsFdU cC

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    Kinetic Energy of a Rigid Body in Plane Motion

    Consider a rigid body of mass m in plane motion.

    ( )2

    212

    21

    22

    2

    12

    2

    1

    2

    212

    21

    Ivm

    mrvm

    vmvmT

    ii

    ii

    +=

    +=

    +=

    Kinetic energy of a rigid body can be separated into:

    - e ne c energy assoc a e w e mo on o

    the mass center G and

    - the kinetic energy associated with the rotation of

    the body about G.

    Consider a rigid body rotating about a fixed axisthrough O.

    ( )

    2

    21

    22

    212

    212

    21

    O

    iiiiii

    I

    mrrmvmT

    =

    ++=

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    Systems of Rigid Bodies

    For problems involving systems consisting of several rigid bodies, the

    principle of work and energy can be applied to each body.

    We may also apply the principle of work and energy to the entire system,

    2211 TUT =+ = arithmetic sum of the kinetic energies of

    all bodies forming the system

    = work of all forces actin on the various

    21,TT

    21U

    bodies, whether these forces are internalor external to the system as a whole.

    For problems involving pin connected members, blocks and pulleys

    connected by inextensible cords, and meshed gears,- internal forces occur in pairs of equal and opposite forces

    - points of application of each pair move through equal distances

    - net work of the internal forces is zero

    - work on the system reduces to the work of the external forces

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    Conservation of Energy

    222

    1222

    12 IvmT +=

    Consider the slender rod of mass m.

    Expressing the work of conservative forces as a

    change in potential energy, the principle of work

    and energy becomes

    2211 VTVT +=+

    =

    =

    +=+

    sin3

    sin2

    1

    32

    10

    222211

    l

    g

    mglml

    VTVT

    ( ) ( ) 2222121212

    21

    21

    321 mlmllm =+=

    sinsin21

    21

    2 mglWlV ==

    mass m

    released with zero velocity

    determine

    at

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    Power

    Power = rate at which work is done

    For a body acted upon by force and moving with velocity ,F

    v

    vFdt

    dU ==Power

    upon by a couple of moment parallel to the axis of rotation,M

    Mdt

    dM

    dt

    dU===Power

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    Sample Problem 17.1

    SOLUTION:

    Consider the system of the

    flywheel and block. The work

    done by the internal forces exertedby the cable cancels.

    Note that the velocity of the block

    For the drum and flywheel,

    The bearing friction is equivalent to acouple of At the instant shown,

    the block is moving downward at 2 m/s.

    .mkg16 2=I

    m.N90

    Determine the velocity of the block after it

    has moved 1.25 m downward.

    Apply the principle of work and

    kinetic energy to develop an

    expression for the final velocity.

    drum and flywheel are related byrv =

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    Sample Problem 17.1

    Apply the principle of work and kinetic energy to develop an

    SOLUTION:

    Consider the system of the flywheel and block. The work

    done by the internal forces exerted by the cable cancels.

    Note that the velocity of the block and the angular velocity of

    the drum and flywheel are related by

    0.4srad5

    m4.0

    sm2 222

    11

    v

    r

    v

    r

    vrv ======

    expression for the final velocity.

    ( )( ) ( )( )

    J440

    srad5mkg162

    1sm2kg120

    2

    1 22

    2

    1212

    121

    1

    =

    +=

    += ImvT

    ( ) ( ) 22

    2

    22

    2

    2

    2212

    221

    2

    110

    4.0

    16

    2

    1120

    2

    1v

    vv

    IvmT

    =

    +=

    +=

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    Sample Problem 17.1

    J44021212

    121

    1 =+= ImvT

    2

    2

    2

    2212

    221

    2 110vIvmT =+=

    Note that the block displacement and pulleyrotation are related by

    rad125.3m4.0

    m25.122 ===

    r

    s

    Principle of work and energy:

    ( ) ( )

    sm85.3

    101J1901J440

    2

    2

    2

    2211

    =

    =+

    =+

    v

    v

    TUT

    sm85.32 =v

    ( ) ( )

    ( )( )( ) ( )( )J1190

    rad125.3mN90m25.1m/s81.9kg1202

    121221

    =

    =

    =

    MssWUThen,

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    Sample Problem 17.2

    SOLUTION:

    Consider a system consisting of the two

    gears. Noting that the gear rotational

    speeds are related, evaluate the finalkinetic energy of the system.

    Apply the principle of work and energy.

    Calculate the number of revolutions

    mm80kg3 ==

    ==

    BB

    AA

    km

    The system is at rest when a moment

    of is applied to gearB.

    Neglecting friction, a) determine thenumber of revolutions of gearB before

    its angular velocity reaches 600 rpm,

    and b) tangential force exerted by gear

    B on gearA.

    mN6 =M

    required for the work of the applied

    moment to equal the final kinetic energy

    of the system.

    Apply the principle of work and energy to

    a system consisting of gearA. With the

    final kinetic energy and number ofrevolutions known, calculate the moment

    and tangential force required for the

    indicated work.

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    Sample Problem 17.2

    SOLUTION:

    Consider a system consisting of the two gears. Noting

    that the gear rotational speeds are related, evaluate the

    final kinetic energy of the system.

    ( )( )

    srad1.25100.0

    8.62

    srad8.62mins60

    revrad2rpm600

    ===

    ==

    BBA

    B

    r

    .

    ( )( )

    ( )( ) 222

    222

    mkg0192.0m080.0kg3

    mkg400.0m200.0kg10

    ===

    ===

    BBB

    AAA

    kmI

    kmI

    ( )( ) ( )( )

    J9.163

    8.620192.01.25400.02

    212

    21

    2212212

    =

    +=

    += BBAA IIT

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