Динамика материальной точки: Учебное пособие

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  • 1

    .

    -

    , - .

    .

    . ..., . .., .., ... 2008.

  • 2

    .

    . -

    . - .

    - , .

    ( ):

    , .

    , , -

    . - . - .

    , .

    . - -, .

    ( ):

    - .

    Fam = (1) , -

    , ,

    = kFam . :

    === .,, kzzkyykxx FmaFmaFma (2) -

    ).(),(),( tzztyytxx ===

  • 3

    .2

    2,, 2

    2

    2

    2

    dt

    zdzyx

    adt

    ydadt

    xda === (2)

    :

    === kzkykx Fdt zdmFdt ydmFdt xdm 22

    2

    2

    2

    2

    ,, , (3)

    x, y, z , Fkx, Fky, Fkz - - .

    .

    . () , ,

    , .

    - )(),(),( tzztyytxx === , (1.3) - , .

    .

    .

    === .,, 222222 kzkykx Fdt zdmFdt ydmFdt xdm

    ,,,,,, 22

    2

    2

    2

    2

    zdtdzVy

    dtdyVx

    dtdxVz

    dtzdy

    dtydx

    dtxd

    zyx &&&&&&&&& =========

    .,, zyx FzmFymFxm === &&&&&&

  • 4

    , ( x, y, z), -, zyx &&& ,, , t, ..

    ).,,,,,,(

    );,,,,,,();,,,,,,(

    tzyxzyxFzm

    tzyxzyxFymtzyxzyxFxm

    z

    y

    x

    &&&&&&&&&&&&&&&

    ===

    (4)

    - -

    , : ).(),(),( tzztyytxx === -

    , -. , - .

    , :

    ).,...,,,(),,...,,,(),,...,,,(

    621

    621

    621

    CCCtzzCCCtyyCCCtxx

    ===

    (5)

    , -

    (5):

    ).,...,,,(

    ),,...,,,(),,...,,,(

    621

    621

    621

    CCCtzzV

    CCCtyyVCCCtxxV

    z

    y

    x

    &&&&&&

    ======

    (6)

    -

    , .. t = 0 - , -, .

    : t = 0, x = x0, y = y0, z = z0 , Vx= 0x& , Vy = ,0y& Vz = 0z& .

  • 5

    (5) (6) , , - 1, 2,6:

    ).,...,,,0(),,...,,,0(),,...,,,0(

    6210

    6210

    6210

    CCCzzCCCyyCCCxx

    ===

    (7)

    ).,...,,,0(

    ),,...,,,0(),,...,,,0(

    6210

    6210

    6210

    CCCzzV

    CCCyyVCCCxxV

    z

    y

    x

    &&&&&&

    ======

    (8)

    (7) (8), -

    , 1,2,,6 - (5) (6) , - , :

    )(),(),( tzztyytxx === , ).(),(),( tVVtVVtVV zzyyxx === 1. , , -

    . 2. ,

    , - . - , , - .

    3. - .

    4. , - .

    5. -.

    6. .

    7. . 8. -

    .

  • 6

    -

    . , , - .

    - -.

    , , . , -: t, , , z

    zyx &&& ,, . : ) , ; ) , ; ) , .

    .

    1. (.1.1) - : ktjitF 2842 = , kji ,, - - .

    kFjFiFF zyx ++= ,

    ,2tFx = ,4=yF .8 2tFz =

    ,222

    tdt

    xdm =

    422

    =dt

    ydm ,

    .8 222

    tdt

    zdm =

    k F

    M (x,y,z)

    x

    y

    z

    x y

    z

    i j

    .1

  • 7

    2. R , (.2).

    VkR = , k- , V - -.

    :

    dtdxkkVR xx == , dt

    dykkVR yy == , dtdzkkVR zz == .

    , R , -

    gm ,

    ,22

    dtdxk

    dtxdm =

    ,22

    dtdyk

    dtydm =

    .22

    dtdzkmg

    dtzdm =

    3. (.2), , -

    R = kV2 , - : VR . R V - , , 2kVR = , - .

    .kmgG = RR = , VV = , VkVkVVkV == 2 . , VkVR = . ,xx VkVR = ,yy kVVR = .zz kVVR = 222 zyx VVVV ++= ,

    .,, zdtdzVy

    dtdyVx

    dtdxV zyx &&& ======

    , :

    k

    R M(x,y,z)

    x

    y

    z

    x y

    z

    i j

    V

    .2

    mg

  • 8

    ,222 xzyzkRx &&&& ++= ,222 yzyxkRy &&&& ++= zzyxkRz &&&& 222 ++= . , c

    =xm && ,222 xzyzk &&&& ++

    =ym && ,222 yzyxk &&&& ++ =zm && zzyxk &&&& 222 ++ - mg.

    , ,

    , .. 0,0 == yx && . :

    =zm && .2 mgzk & 4. (.1.3) -

    Q , , , .. 2r

    fQ = , rQ , r - -, , f -.

    Q

    erfQ 2= ,

    e - r . r,

    errfQ 3= .

    rer = , .3 rrfQ =

    ,33 xrfr

    rfQ xx ==

    k

    Q

    M(x,y,z)

    x

    y

    z

    x y

    z

    i j

    .3

    r

    e O

  • 9

    ,33 yrfr

    rfQ yy ==

    zrfr

    rfQ zz 33 == .

    - 21222222 )( zyxzyxr ++=++= .

    Q

    2

    3222 )( zyx

    fxQx ++= ,

    23222 )( zyx

    fyQy ++= ,

    23222 )( zyx

    fzQz ++= .

    ,)( 2

    3222 zyx

    fxxm++

    =&& ,)( 2

    3222 zyx

    fyym++

    =&& 2

    3222 )( zyx

    fzzm++

    =&&

    , .

    - .

    , , . -

    F.

    constFdt

    xdm ==22

    ,

    dt

    dVdt

    xd x=22

    ,

    Fdt

    dVm x = .

    dt, .. FdtdVm x = .

  • 10

    .1CtFmVx += t = 0, Vx = V0, 1 =mV0.

    , 0mVtFmVx += .

    dtdxVx = ,

    0mVtFdtdxm += .

    , dt: dtmVtFmdx 0+= .

    102

    2CtmVtFxm ++= .

    : t = 0, x = x0, 2 = mx0.

    ,

    .2

    2

    00 mFttVxx ++=

    ,

    . .

    (.4) -

    , f, m, V0. -.

    - , - . = 0.

    gm , - N , F .

    -

    Fmgdt

    xdm = sin22

    .

    mg

    N F

    y

    x

    O

    .4

  • 11

    F = fN, cosmgN = , cosfmgF = .

    m

    cossin22

    fggdt

    xd = .

    22

    dtxd

    dtdVx ,

    cossin fggdt

    dVx = . , dt:

    dtfggdVx )cossin( = , 1)cos(sin tfgVx += . t = 0, Vx =V0,

    1= V0. , 0)cos(sin VtfgVx += . xV = dt

    dx ,

    0)cos(sin Vtfgdtdx += .

    dt: dtVtdtfgdx 0)cos(sin += .

    202

    2)cos(sin CtVtfgx ++= .

    t = 0, x0 = 0, 2= 0. ,

    tVtfgx 0

    2

    2)cos(sin += .

  • 12

    , . )(tFF = ,

    0V . .

    )(22

    tFdt

    xdm x= . -

    dt

    dVdt

    xd x=22

    ,

    xV :

    ).(tFdtdV

    m xx =

    dt, : .)(