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, : .)( dttFmdV xx =

    += 1)(1 CdttFmV xx .

    dtdxVx =

    .

    11

    mdt

    dx += ,

    = 1)( dttF x . :

    dtCmdx )1( 1+= ,

    -

    ++= 211 CtCdtmx .

  • 13

    1 2 xV

    . .

    gm (. 1.5) - VkR = .

    z, , , t = 0, z0 = 0, 00 =z& .

    zkVmgdtzdm =2

    2

    .

    dt

    dVdt

    zd z=22

    , m , k/m = n,

    zz nVgdtdV = .

    , ,

    dt ( zkVg ), -

    dtnVg

    dV

    z

    z = .

    1dtVgndV

    z

    z += .

    1)ln(1 CtnVgn z

    += .

    :t = 0, Vz = 0, , gn ln11 = .

    gk

    tnVgn z

    ln1)ln(1 = . :

    R

    mg

    O

    z

    z

    .5

  • 14

    ntgnVg z = ln)(ln( , ntgnVg z =ln , ntz eg

    nVg = . ,

    )1(tne

    ngVz

    = .

    , t Vz = g/n =mg/k .

    , , dtdzVz = ,

    - z:

    )1( nteng

    dtdz = .

    dt,

    dtengdz nt )1( = ,

    2)1( Cdtengdz nt += .

    2)( Cnet

    ngz

    nt++=

    .

    t=0, z = 0 C2= 2n

    g .

    ))1(1( / += mkten

    tngz .

    .

    (. 1.5) : gm VkVR = , k -. V0 = 0, z - , , z0 = 0. .

    z

  • 15

    222

    zVkmgdtzdm = .

    dt

    dVdt

    zd z=22

    , 2zz VkmgdtdVm = .

    m ,

    dz:

    dzdVV

    dzdtdzdV zzz = ,

    dtdzVz = , 2zzz Vm

    kgdzdVV = ,

    , dz

    ( 2znVg ), mkn = :

    dznVgdVV

    z

    zz = 2 . :

    12 CdzVngdVV

    z

    zz += ; 12 )ln(1 CznVgn z += . z = 0, Vz = 0,

    gn

    C ln11 = .

    znVgg

    n z= 2ln

    1, ).1( nzz en

    gV =

    . z nt -, z . , z , .

    kmg

    ngV == .

    , .

    ,

    (.6), P , -

  • 16

    , , P :

    MOmkP 2= . x= k2m x. -

    , .. t=0, x0=a, Vx = 0.

    , , gm N , .

    :

    mxkdt

    xdm 222

    = . m: 02 = xkx&&

    . - :

    022 = kr . kr =2,1 .

    , -,

    ktkt eCeCx += 21 . () 1 2

    . (), : ktkt keCkeCx = 21& . () () () t= 0, x0=a, 00 =x& 21 CCa += , 0 =C1 - C2. 1 = 2 = 2

    a.

    1 2 () () :

    )(2ktkt eeax += , )(2

    ktkt eeakxVx== & .

    N

    mg

    x x

    O

    .6

  • 17

    . , F= - f N, f -

    , N - .

    fNxmkxm = 2&& . N = f mq (.7), fmgxmkxm = 2&& . m fqxkx =+ 2&& . () -

    1 02 =+ xkx&& 2 (), .. 21 xxx += .

    ktkt eCeCx += 211 . 2 (), .. -

    : Ax =2 , 02 =x&& . 2 (), -

    fgAk =+ 20 , 2kfgA =

    221 kfgeCeCx ktkt ++= . ()

    1 2 ktkt keCkeCx = 21& . () () () t =0, x0 = a, 00 =x& ,

    221 kfgCCa ++= , kCkC 210 = .

    )(21

    221 kfgaCC == .

    1 2 () ()

    22 ))((21

    kfgee

    kfgax ktkt ++= , ).)((2 2

    ktkt eekfgakxVx

    == &

    N

    mg

    x x

    O

    .7

    F

  • 18

    .

    , = 0kzF , .0,0 0 == zz & - . t =0, x = x0, y = y0 , 00 , yyxx &&&& == .

    :

    .

    ,

    2

    2

    2

    2

    =

    =

    ky

    kx

    Fdt

    ydm

    Fdt

    xdm (9)

    ,

    , , .. . - 1, 2, 3 4, . , - x = x(t), y = y(t).

    - : , , , , - .

    , . m 0V ,

    , - .

    , , ,

    0V (.8).

    - gm .

    :

    mgdt

    ydmdt

    xdm == 22

    2

    2

    ,0 .

    V0

    mg

    .8

  • 19

    : dt

    dVdt

    yddt

    dVdt

    xd yx == 22

    2

    2

    , .

    m,

    .,0 gdt

    dVdt

    dV yx == dt , Vx = C1, Vy= - gt+C2. () t. t = 0, xo = 0, Vx = Vo cos , Vy =Vosin . , 1= Vo cos , 2= Vo cos . (),

    dtdyV

    dtdxV yx == ,

    .sin,cos 00 gtVdtdyV

    dtdx == . ()

    -

    : .sin,cos 00 gtVVVV yx == .)sin(cos 2022022 gtVVVVV yx +=+= () (), x = V0 t cos + C3, y = V0 t sin 0,5 gt2 + C4. 0 = 0, 0 = 0, ,

    1 = 2 = = 0. :

    x = V0 t cos, y = V0 t sin 0,5 gt2. -

    .

  • 20

    1. . () t, t :

    ,cos0 Vxt = 222

    0 cos2x

    Vgxtg = . ()

    () . 2. . ,

    = 0.

    0cos2

    222

    0

    = xV

    gxtg , 1 = 0, tg

    gVx

    220

    2cos2= .

    1 , 2 - , = L

    L = gV 2sin20 .

    , sin2 =1, .. = 450.

    3. , -

    g

    VLx O 2sin2/2

    == :

    g

    VH O2sin22 =

    4. . x L

    x = V0 t cos ,

    gVT sin2 0= .

    5. ().

    , , , - .

    , = 450 , VO = 28,3 /c, L 81,6 ; H 20,4 ; T 4 c. Microsoft Excel (.9).

  • 21

    ,

    2 VO = 8,3

    /c 450 . - , -, = 0,5.

    , - ,

    , , , , - , 0V (.1.10).

    . - , t = 0, xo = 0, yo = 0.

    ./20707,03,2845sin

    ;/20707,03,2845cos0

    0

    cVyVcVxV

    OOOy

    OOOx

    ======

    &&

    t = 0, ./20,/20 cycx OO == &&

    .

    0

    5

    10

    15

    20

    25

    0 10 20 30 40 50 60 70 80

    . 9

    x

    y

    O

    R

    V

    mg VO 450

    . 1.10

  • 22

    VR = , .

    .

    ,

    dtdyVR

    dtdxVR

    yy

    xx

    ==

    ==

    mg .

    .

    ;

    2

    2

    2

    2

    mgdtdy

    dtydm

    dtdx

    dtxdm

    =

    =

    -

    :

    ;025,022

    =+dtdx

    dtxd

    ()

    8,925,022

    =+dtdy

    dtyd...

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