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					   Dynamique du point matériel                                     168

Corrigés des exercices de 5.1 à 5.20                                                                    20.5                 1.5

                                                                                                                                                   :1.5
                                     v= r :                                                                                                                       /
                                10.6, 28
                              =            1, 05rad .s 1 :                                                                                                    !
                                   60
                              : EE '                      #$% &                                         '                                       ( )*! "       !
                                 r = l.sin 60° , r = 4, 5.0,87                                          r = 3, 9m
                                             v = 1, 05.3,9                   v          4,1ms               1
                                                                                                                                           :!
 2 (   /0'      ' 1'!                              %   &                 :                                 +                ,   - ( - . "# /"
             .( . 1!5)                                 2 &             & ! .m                   2
                                                                                                         #$' .                - ( #$' *
                                           E
                                                                                                y'
                                                   60° =                                Ry
                                                                    T
                                                                                                Ty              R

                                       x                                    Tx                                             x'
                                                                                                D               Rx



                                            E'                                                      y
                                           P +T + R = m                 2
                                                                            r.i
                                           T .sin              R.cos            =m          2
                                                                                                (1)
                                                                                                r
                                           P R.sin                T .cos                = 0 ( 2)
                                                                                        : ( 2 ) (1)                         ' #,            %#         ' ' )7 !
                          T .sin           R.cos + m               2
                                                                       r                                R.cos + m                2
                                                                                                                                     r
                                       =                                            tg =
                          T .cos             P R.sin                                                      P R.sin
                                   (
                          R = m g.sin                      2
                                                               .r.cos        ) ( 3)                         ; R           37 N

                                                                             : ( 2 ) 5 (1)                           #,                  '!' !            ' ' /8
                                          R.cos + m                     2
                                                                            r
                                       T=                                               T               46, 4 N
                                               sin
                                          P R.sin
                                       T=                                       T       43.42 N
                                             cos
                                                                 .# 7 0! ' % &' &                                                    9'#!        ' &      %: ;
        : ( 3)   #,                '!' !                          2 '     ,    ,!                                                     <                          /
                          (
                 R = m g .sin                  2
                                                   .r.cos        )=0
                          g .sin              g .sin                                        g
                  2
                      =                =                                            =                            ,          2,1rad .s       1

                          r.cos            l.sin .cos                                    l cos




A.FIZAZI                                               Univ-BECHAR                                                                                LMD1/SM_ST
     Dynamique du point matériel                           169

                                                                                              :2.5
                       R                                              '.          - /0 2 &        / ! /1
                       O                                             , = % >           # 7 :&
           l1                        l2
A                                                     B           % ! #% " ?&       &%   2 &
T1                                                                T/
                                                                      = T / : =5 # , (
                                                                                  1
                                                                                             )
                                                      T
     T1                                                          7       5  . T ' ' - . " ! 5 #!
                       F                          T
                                                                  % ! #% m3  m2    '' > # ' @ 5 " !
 m1
                                            T2            T3      (<, : % '% "# !      %    ' ' - %
P1                                                                                       :A '    # B
                                                          T3
                                                      m3                  P3 T3 = m3 .a                     m3 m2
                                                                                                     a=             g
                                             T2                            P2 + T2 = m2 .a                  m2 + m3
                                                          P3
                                               m2
                                            P2
                                                                                                                               :!
                   P3 T3 = m3 .a          T3 = m3 ( g a )
                       P2 + T2 = m2 .a     T2 = m2 ( g + a )
                                                                                                m2 .m3
                     m m2                                                             T = 4g
                   a= 3      g                                                                 m2 + m3
                     m2 + m3
                   T = T2 + T3 , T = m2 ( g + a ) + m3 ( g a )
                                                                                                    P = T1 : m1 C
                                                                                                     1                    % ! #%
                                                                 T/
                                                                      =    T1 /
                                                                                       T1.l1 = T .l2 :          ,      1! "
                                                     m2 .m3
                                   m1 g .l1 = 4 g           .l2        m1 ( m2 + m3 ) .l1 = 4m2 m3 .l2 :                   B
                                                    m2 + m3
                : T1       T   '    '       ' &       *        = # ' " ?&                                #$&%         ' - & /2
                                                                          4m2 m3
                                   R = T1 + T          R = g m1 +
                                                                          m2 + m3


                                                                                                                     3.5
                                                                                        ( ; 5 . 1!5):
                                                               . % !                  #%     & A '    ' #5 !
                                           : m3       m2 D m1                         A '       # B % : % '% 5 %!
                                   T1 = m1a1
                                   P3 + T3 = m3 .a3            T1 = m1a1

                                   P2 + T2 = m2 .a2            2 P3 + T1 = 2m3 .a3
                                            1                  2 P2 + T1 = 2m2 .a2
                                   T2 = T3 = T1
                                            2




A.FIZAZI                                         Univ-BECHAR                                                        LMD1/SM_ST
   Dynamique du point matériel                                 170

- % > # '= #                    > # ' . aa = ar + ae : 5 (        %) %# !@ % !                  ,!
 .a     A '.                 $ m3 m2 ' '       % ! > # ' # 5 .( ae = a1 ) m1 ' > # ' =5       '
                                                                   : .           +?   E# '@ "
                                                  a2 = ar a1 : :         #$ # '     m2 '    % ! #%
                                                  a3 = ar + a1 : :       #$ # '     m3 '    % ! #%
                                                                            : %#' F #!!    #& G#%
                                                T1 = ma1      (1)
                                                T1 2 P2 = 2m2 ( ar             ( 2)
                                                                            a1 )
                                                   T1 + 2 P3 = 2m3 ( ar + a1 ) ( 3)
                                                                        .          # H</ 7                 @ #,     #!  ! '
                                                                    : ( 3)         #,  A '.                 % ! > # ' 8 ' !
                                                    2m3 g 2m3a1 m1a1
                                            ar =                     g                  ( 4)
                                                          2m3
                                            : m1      '     a1 > # ' - #%               ! ( 2)        #,                #$' &% a I ,!
                                                          4m2 m3
                                            a1 =                        g                ( 5)
                                                   m1m2 + m1m3 + 4m2 m3



                     +                                                 R1          a1
                                                                                        T1 T1
                                                                            m1
                                                                                                               +
                                                                              P1
                                                                                                               T
                             T                                                                                     +
                     m1                                                                                   T
            a1                                                                                                            ae = a1
                P                                                                                    T2            T3
                 1
                          + T
                                      ae = a1                                                                      T3
                     T2          T3                                                                            m3        a3
                                 T3                                                                  T2
                             m3        a3                                                                          P3
                     T2                                                                         a2        m2
                                 P3
                                                                                                     P2
                 a2 m2
                     P2




 : ( 5)    #,             # #!          '    ' &% :           > # ' I ,!                 ( 4 ) % ! > # ' - #%                       ,!
                                                      m3 m1 m1m2
                                            ar =                        g                ( 6)
                                                   m1m2 + m1m3 + 4m2 m3




A.FIZAZI                                                  Univ-BECHAR                                                     LMD1/SM_ST
    Dynamique du point matériel                             171

                                                    . a3      a2       &%'         # ' 8#'!' $       F #%
                                                                                      : m2 ' a2 > # ' - #%
                                             m3m1 m1m2                               4m2 m3
                   a2 = ar    a1 ; a2 =                        g                                   g
                                          m1m2 + m1m3 + 4m2 m3                m1m2 + m1m3 + 4m2 m3
                                                    m3m1 m1m2 4m2 m3
                                             a2 =                        g
                                                    m1m2 + m1m3 + 4m2 m3
                                                                                         : m3 '      a3 > # ' - #%
                                             m3m1 m1m2                   4m2 m3
                   a3 = ar + a1 ; a3 =                         g+                      g
                                          m1m2 + m1m3 + 4m2 m3    m1m2 + m1m3 + 4m2 m3
                                                    m3m1 m1m2 + 4m2 m3
                                             a3 =                        g
                                                    m1m2 + m1m3 + 4m2 m3


                                                                                  (E< 5          . 1!5):
                                               : m3     m2 D m1                  A '             # B % : % '% 5 %!
                                      T1 = m1a1
                                      P3 + T3 = m3 .a3         T1 = m1a1

                                      P2 + T2 = m2 .a2         2 P3 + T1 = 2m3 .a3
                                               1               2 P2 + T1 = 2m2 .a2
                                      T2 = T3 = T1
                                               2
    =5         '     - % > # '= #  > # ' . aa = ar + ae                                  B #                #! .5 #
           .a            A '. $ m3 m2 ' '       % ! > # '                             # 5 .( ae = a1 ) m1 ' > # '
                                                        :                            .         +?        E# '@ "
                                     a2 = ar a1 : :                                   #$ # '          m2 '      % ! #%
                                     a3 = ar + a1 : :                                  #$ # '          m3 '     % ! #%
                                                                                          : %#' F #!!          #& G#%
                                          T1 = ma1      (8)
                                          T1 2 P2 = 2m2 ( ar a1 ) ( 9 )
                                            T1 + 2 P3 = 2m3 ( ar + a1 ) (10 )
                                                                   .         # H</ 7             @ #,     #!  ! '
                                                               : (9)         #,  A '.             % ! > # ' 8 ' !
                                        ( m1    2m2 ) g     ( m1 + 2m2 ) a1 g
                                 ar =                                                (11)
                                                        2m2
                                      : m1     '      a1 > # ' - #%            ! (10 )      #,      #$' &% a I ,!
                                             4m2 m3 m1m2 m1m3
                                      a1 =                        g              (12 )
                                             m1m2 + m1m3 + 4m2 m3
: (12 )   #,           # #!       '     ' &% :             > # ' I ,!           (11) % ! > # ' - #%                ,!




A.FIZAZI                                            Univ-BECHAR                                        LMD1/SM_ST
   Dynamique du point matériel                                172

                                            2m3m1 2m1m2
                                 ar =                          g                        (13)
                                          m1m2 + m1m3 + 4m2 m3
                                                                  .        &%'            # ' 8#'!' $       F #%
                                                                                             : m2 ' a2 > # ' - #%
                                        2m3m1 2m1m2                                  4m2 m3 m1m2 m1m3
             a2 = ar    a1 ; a2 =                          g                                              g
                                      m1m2 + m1m3 + 4m2 m3                           m1m2 + m1m3 + 4m2 m3
                                                   3m3m1 m1m2 4m2 m3
                                          a2 =                           g
                                                    m1m2 + m1m3 + 4m2 m3
                                                                                                 : m3 '             a3 > # ' - #%
                                        2m3m1 2m1m2           4m2 m3 m1m2 m1m3
             a3 = ar + a1 ; a3 =                           g+                      g
                                      m1m2 + m1m3 + 4m2 m3    m1m2 + m1m3 + 4m2 m3
                                                   4m2 m3 m1m3 3m1m2
                                          a3 =                          g
                                                   m1m2 + m1m3 + 4m2 m3


                                                                                                                        4.5
                                                                            :     J&        <               /
A# ' @              ' '          0   ><(                    5            1 B #$' ( !      A# ' @ - ( K %' #
                                       :                J & #$!       D Px &/ %    J 0 # ''
               f s ,max = Px = mg sin      0

               f s ,max = µ N                       tg    0   = µ , tg           0   = 0,80           0   = 38, 66°
               N = Py = mg cos        0

                                                                                              : 1 B A# ' @ - ( - . /"
                                       f s ,max = µ N , f s ,max = 3,13N
                                                                                              : 35°             !      1#! - & /8
                                     N = Py = mg cos                  , N = 4,1N
                                                                                         : 35° !            !         A# ' @ - ( /
                                     f s = Px = mg sin              , f s = 2,87 N

                                                                           y'
                                     x'
                                                                      N
                                                    fs
                                                                    m
                                                                      Px
                                                   Py
                                                              P                           x

                                               y




A.FIZAZI                                           Univ-BECHAR                                                        LMD1/SM_ST
   Dynamique du point matériel                                        173

                                                                                                                                           5.5
           :       , 2 & >                     5 !, 7 D '%#/                                 %               &'!                    <                   /
                                                             fc + P + N = 0
                                                                                                                        :                         #& G#%
            Px      fc = 0       mg sin       0   = µc N
                                                                          tg       = µc , tg            = 0, 40 ,               = 21,8°
            Py     N =0          N = mg cos             0
                                                                               0                    0                       0


                                                                                                        : 35°                   !         1#! - & /"
                                              N = mg cos                      , N = 6,55 N
                                                                                                     : 35° !                            A# ' @ - ( /
                                                  f c = µc N ; f c = 2, 62 N
                                                                                                                     : 35°                ! > # ' /E
                                                                      mg sin                 fc
                             mg sin           f c = ma                                            , a = 2, 46 N
                                                                           m


                                                                                        y'
                                         x'
                                                                                   N
                                                             fc
                                                                               m
                                                                                   Px               +
                                                            Py
                                                                          P                              x

                                                    y

                                                                                                                                            6.5
M;!    #' #%      M;!                                                 5            #,       L#&% J                                    :@ !      . /
:       > # ' = #   5 "                           % !                                1!   ). %#/ 2 '                                   % ! #% > # '
                                                                                            .( A                                      > # ' B
.(5) . .#,                    L#&% J                         : !'                   A        #$& % ' "                                  - & NF     '
                                                              :                     > # '" ! A '       #                              B (<, :% !
                                                                                                                                     :A        % ! #%
                                                                                                                F
                 F + P + N = ( mA + mB ) .a , F = ( mA + mB ) .a                                        a=                          (1)
                        0                                                                                    mA + mB
A# ' @ - ( 7          .A               % ! #%           #          !5 @ %#/                  ,       % ! #% '     O :B                              % ! #%
                                                                  :"' ! 5                        . !       A# ' @ - (                             - /P
                              f s ,max, A = mA .a
                                                                          µ s mA g
                            f s ,max, A = µ s N A                a=                              a = µs g          ( 2)
                                                                              mA
                            N A = PA = mA g
                                                  : ( 2)           (1) ' #,                       %- #              ;           F - & - . 8#'!' @




A.FIZAZI                                                    Univ-BECHAR                                                                   LMD1/SM_ST
     Dynamique du point matériel                                         174

                               F
                    a=              = µs g                          F = µ s ( mA + mB ) g , F = 15, 7 N
                            mA + mB


                                               N                                                  NB N
                                          B                              +         fs,B            B                        +
                                  A                                  F                         A                            F
                                                                                                PB

                                               P                                                       P


                                                                                     :F - & : % ' !      > # ' /"
      (") . . F         P, N 2 &               ,?#                           7 . ##'      ' @ :@ !@ 2 '   % ! #%
                                                                               : %#' #!! A '        # B (<, : % '
                            P+N =0                                            F
                                                                 a=                  , a = 1,96ms           2

                            F = ( mA + mB ) a                            ( mA + mB )

BS        .(8    . ) A QR                             &%         - &  !# 7 D A            % ! #% D B N   > # ' /8
     (A         % ! #%                    B                    B        A# ' @ - () f c, B PB , N B 2 ( H</ J?#
                  :B                       A          '          # B (<, :% ! . F - & J? =7 A
                                               NB                                                      NB
                    fc, B            B                           F                    fc,B        B                     +
                                                                  +                                                         F
                                 A                                                            A

                                              PB                                                       PB

                   PB + N B = 0
                     f c , B = mB a '                      µc mB g
                                                   a' =                       a ' = µc g , a ' = 0,98ms             2

                   f c , B = µc N B                            mB
                   N B = mB g
                                      .               !      "                #       $%        B
2 ( J% B J?#                B             .(          . ) ;!                           &%      - &  !# 7 B              > # '
                                                     :B                      A    '          # B (<, :% ! . f c, B PB , N B , F
                       P+N =0
                       F f c , B = mB a "                             F µc mB g
                                                               a" =             , a " = +0, 98ms                2

                       f c , B = µc N B                                 mB
                       N B = mB g
                                                           .             "        & $%         B




A.FIZAZI                                                       Univ-BECHAR                                                  LMD1/SM_ST
   Dynamique du point matériel                                           175

                                                                                                                                              :7.5
                                                                 : ''                                A            '      # B (<, :% !
                f2                   N2                                             P + N1 + N 2 + f1 + f 2 = m1a1
                                                                                     1
           f1
                                m2        N1                                        P2 + N 2 + f 2 = m2 a2
                         m1
                                                                        : #        2 '               =                                        ' #%,   & !
                                      P2                                          m1 g sin               f1        f 2 = m1a1       (1)
                           P1                                                     m2 g sin       1        f 2 = m1a2            ( 2)
                                                                                                              :              A# ' @           ' (      %,!
                                     f1 = h1 ( N1 + N 2 )
                                     N1 = m1 g cos                     f1 = h1 g ( m1 cos + m2 cos                       )
                                     N 2 = m2 g cos
                                     f 2 = h2 N 2
                                                                      f 2 = h2 m2 g cos
                                     N 2 = m2 g cos
        : '                     ' #,                  * ! ( 2)                (1)       ' #,                                  A# ' @           ' ( I ,!
                     m1 g sin              m2 gh2 cos                h1 g ( m1 cos + m2 cos                       ) = m1a1       ( 3)
                     m2 g sin          1       h2 m2 g cos       = m2 a2 ( 4 )
                                                                              : ( 4 ) ( 3) ' #,                                         # '     F 9'!' !
                                                                     m2
                     a1 = g ( sin                h1 cos      )          g cos       ( h2 + h1 )                   a1 = 3,53ms       2

                                                                     m1

                     a2 = g ( sin                 h2 cos     )          a2 = 7, 79ms         2




                                                                                                                                              8.5
   L % =5                     ><(               . .( )                        %          #                                   - /P        2 &           /!
                                                                                                          : T = PB            T = f s ,max
                         T = f s ,max
                         T = PB = mB g
                                                                       mC =
                                                                              ( mB µs mA )                    , mC = 15kg
                          f s ,max = µ s N                                             µs
                         N = PA = ( mA + mC ) g

           A         '           # B (<, : % '% > # '                                    * ! #!! ($                             )C              J !
                                                                                                                                                  :
                                                                 T     f c = mA a
                                                                 PB T = mB a                 a=
                                                                                                         ( mB µc mA ) g                 , a = 1.36ms   2

                                                                                                                  mB + mA
                                                                 f c = µc mA g




A.FIZAZI                                                     Univ-BECHAR                                                                  LMD1/SM_ST
   Dynamique du point matériel                                         176

                         N                                                                      N
                                                                                                C
           f s ,max                     T              T                                                             T           T
                        A                                                        f s ,max      A

                                                                       T                                                                        T
                                PA                                                                     PA + PC
                                                                       T                                                                        T

                                                                   B                                                                        B

                                                                       PB                                                                       PB


                                                                                                                                      9.5
                                                                                                                          :) *        &          /I
#$ J? ' ' -                   - & .A              '        # B (<, :% ! /                          .           #$ / ! D2 & * ! /1
                                                                                               :             . P #$ &/     #   &!
                                               F = P = ma
                                                                              a = g = g .u z
                                         P = mg
                                                                                                      v = vx + vz 1                                   /2
                                                                                 : 1'!             &'           :X C                                 :
                                              Fx = 0           vx = v0 x = v0 .cos              (1)
                                                                            : #1'!#% - T'              &'                   :Z C                     :
                                              Fz = P = mg                    az = g = Cte
                                       vz = gt + v0 y = gt + v0 sin                            ( 2)
                                                                                                       : 7                1                     >#,.
                      v = vx .u x + vz .u z           v = v0 .cos .u x + ( gt + v0 sin                  ) .u z           ( 3)

                                                        : OM ( t ) J?                   >#,.                * ! ( 3) - #%,                      # ! /3

                                        OM                     t
                         dOM
                      v=                       dOM =               v0 .cos .u x + ( gt + v0 sin                  ) .uz    dt
                          dt              0                    0



                                                                            1 2
                             OM = v0 .cos .t .u x +                           gt + v0 sin .t .u z                ( 4)
                                                                            2
                                              x
                                                                                   z




      : ( z = 0 ) #$ 5            '     1                  %           "       ! . ( z = 0)        ,        ,! # #                   ; 7& K %' /4




A.FIZAZI                                                   Univ-BECHAR                                                           LMD1/SM_ST
     Dynamique du point matériel                               177

                                                                                0
                                     1 2
                                       gt + v0 sin .t = 0                   t = 2v0 sin
                                     2
                                                                                    g
                                                                     :2                ! x /                G     #,                    I ,!
                                                            2                                          2
                                                          2v0 .sin .cos                               v0 .sin 2
                     x = v0 .cos .t            xmax =                               , xmax =
                                                                  g                                        g
E7      ,!   1           H %! . vz             (#.                        ,!' #        zmax          1 B #$ #;' ; 7& K %' /5
                                                                                                         : ( 2 ) #,
                                                                                    v0 sin
                                     vz = gt + v0 sin = 0                      t=
                                                                                        g
                                                              : ! ( 4)            #,                 z - #%                        F I ,!
                                                                   2
                                                                  v0 .sin 2
                                                      zmax =
                                                                      2g
                                                                                                                   :: ;< => =             /II
                                                              P + f = ma : ' &                     ,?# ; 7& L                      7     /1
                                                                                                  : ?#;'   #,                      :& '! /2
                                     P + f = ma
                                                               dv k
                                         dv                      + v=g                        ( 5)
                                     a=                        dt m
                                         dt
                       : ?#;'             #,          % - .#% v ( t ) 1                                          #,. - #%, 9'!' ! /3
                                                                                                      k
                                                                                                        t        m
                                                                                        v = Ae        m
                                                                                                            +g      :     #$        5 :& '!
                                                                                                                 k
:      !     D t = 0, v = v0 :        '               '%G           .                         * ! =7               A    %#/             ' &%
                                                              m                            m
                                      v0 = Ae 0 + g                       A = v0       g
                                                              k                            k
                                                                                                                               :
                                                                    k
                                                          m           t        m
                                      v = v0            g   e       m
                                                                          +g               ( 6)
                                                          k                    k
                            m
                   vL = g     :           ( 6 ) #,                   * ! D                                       P #                       &
                            k
                                                                    :             * ! (6)                   #,                      &          !
                                               k                                               k
                                     m           t        m                                      t
                     v = v0      g     e       m
                                                     +g             v = ( v0 vL ) e            m
                                                                                                     + vL        (7)
                                     k                    k
                                                      : uz     ux                   #,. @ %                        >#,.             F     %,!




A.FIZAZI                                             Univ-BECHAR                                                          LMD1/SM_ST
   Dynamique du point matériel                                                   178

                              v0 = v0 cos .u x + v0 sin .u z
                                   m
                              vL = g                                           m                             m
                                   k                       vL = g                uz        vL = g
                                                                               k                             k
                              g = gu z
                                                                                                                  k
                                                                                                                    t
                              v=      ( v0 cos                 .u x + v0 sin .u z ) + vL .uz e                    m
                                                                                                                        vL .u z

                                                                  k                                                        k
                                                                    t                                                        t
                              v = ( v0 cos                 )e     m
                                                                        u x + ! vL + ( v0 sin + vL ) e                     m     " uz
                                                                              !                                                  "
                                                     vx                                                vz

                                     :                         ( 7 ) - #%,             #           ; J?                   >#,. - #%                            *       /4

                                                   k
          dOM                                        t
       v=     = ( v0 vL ) e                        m
                                                          + vL
           dt
       OM               t                            k                                                                              k
                                                       t                                                          m                   t
            dOM = !( v0 vL ) e                       m
                                                           + vL " dt                  OM = ( v0              vL )   1 e             m
                                                                                                                                          + vL .t         (8)
       0                0
                                                                                                                  k
                                                         k                 t
                                          k                t
       OM = !( v0 vL )                      e            m
                                                               + vL .t "
                                          m                                0

- #%        #! , #          #$' &% vL                #$ '%              C% v0 I ,!                      #,   .!! OM '%                                             *
                                                                                                       : '#!   #, 1!! /                               1
                                                                                                                  k
                                                                                              m                     t
              OM =          ( v0 cos       .u x + v0 sin .u z ) + vL .uz                        1 e               m
                                                                                                                          vLt.u z
                                                                                              k
                                                                  k                                                                        k
                              m                                     t                                    m                                   t
              OM = ( v0 cos )   1 e                               m
                                                                         u x + ! vL .t + ( v0 sin + vL )   1 e                             m
                                                                                                                                                  " uz
                              k                                                !                         k                                        "
                                                                                                                                          : '%                     *!
                                                         k                                                                        k
                      m                                    t                          m                                             t
           x (t ) =     v0 cos            1 e            m
                                                                   , z (t ) =           ( v0 sin + vL ) 1 e                       m
                                                                                                                                          vL .t          (9)
                      k                                                               k


  ,!' '       1                      %               H %! .                    (#.                          ,!'                  1 B #                   ; 7& K %' /5
                                                                                                                                                  :           E7 #$
                                                                                              k
                                                                                                ts
                                       vz = vL + ( v0 sin + vL ) e                            m
                                                                                                       =0
                                            k                                               k
                                              ts                 vL                           ts             vL
                                       e    m
                                                    =                                  e    m
                                                                                                   =
                                                           v0 sin + vL                                 v0 sin + vL
                                   k
                                     ts                         vL                         k               vL
                            ln e   m
                                           = ln                                              ts = ln
                                                          v0 sin + vL                      m         v0 sin + vL

                                                                         k       v
                                                                 ts =      ln 1 + 0 sin
                                                                         m       vL




A.FIZAZI                                                                Univ-BECHAR                                                                   LMD1/SM_ST
   Dynamique du point matériel                                  179

                                     :# #!               '         & #%                I ,!         (9) ' !            ' #,     J      !
                                               km      v
                                                  ln 1+ 0 sin
                    m                          mk      vL
                xs = v0 cos            1 e
                    k


                                              m                                 1
                                     xs =       v0 cos             1
                                              k                               v0
                                                                         1+      sin
                                                                              vL
                                                             km      v
                                                                ln 1+ 0 sin
                    m                                                                         m       v
                zs = ( v0 sin + vL ) 1 e
                                                             mk      vL
                                                                                       vL .     ln 1 + 0 sin
                    k                                                                         k       vL


                       m                        1                                       m       v
                zs =     ( v0 sin + vL ) 1    v
                                                                                 vL .     ln 1 + 0 sin
                       k                   1 + 0 sin                                    k       vL
                                              vL
                       m                   v0 sin                                    m       v
                zs =     ( v0 sin + vL )                                      vL .     ln 1 + 0 sin
                       k                 vL + v0 sin                                 k       vL

                                              m                           m       v
                                       zs =     v0 sin              vL      ln 1 + 0 sin
                                              k                           k       vL
                                                   k                                                           k
                            m                        t                         m                                 t
                 x (t ) =     v0 cos     1 e       m
                                                               , z (t ) =        ( v0 sin + vL ) 1 e           m
                                                                                                                       vL .t = ( 8)
                            k                                                  k


                                              :t                    z (t )           x (t )                      (9)                  /6
                              m
                 x ( t )t    =  v0 cos = A x ( t )t                           =A          (10 )
                              k
                              m
                 z ( t )t    = ( v0 sin + vL ) vL .t                       z ( t )t       = vL .t + B     (11)
                              k
                                         B

                                                !"             #                t                       (11)
                                                                                         . (10 )                     $% &


                                 .                       &#                                           )#* :                         III




A.FIZAZI                                           Univ-BECHAR                                                       LMD1/SM_ST
      Dynamique du point matériel                                180

 z                                                           z

                                         m
                                xL =       v0 cos
                                         k

  O                                                 x     O                                                                 x




                                                                                                                   10.5
                                                                                              :+   ,
- (                     N-  +      ,  - ( D P &/                                       2 ( H</ ,?#                           /1
        A     '      # B (<, : % '% U ; B    U(5) .                                     #(< ! . f                       A# ' @
                                                                                                                       :"' ! 5
                                                    dv
                                    P + N + f = ma      = P+N + f  m
                                                     dt
                                                          : MN MT                                                     2 &        & !
                                        dv                    dv
                       PT f = maT = m         mg sin     f =m       (1)
                                        dt                    dt
                                           v2                     v2
                         N + PN = maN = m        N + mg cos = m         ( 2)
                                           R                      R
                                                                                       :                   A# ' @ - ( - #%
                           f = µN
                                                                                               v2
                                                        v2        f = µ mg cos             m
                           N = mg cos               m                                          R
                                                        R
                                               :                 ?#;'        #,            * ! (1)           #,             I ,!
                                                                             v2    dv
                                    mg sin          µ mg cos             m      =m
                                                                             R     dt
                                    dv     µ
                                               v 2 = g ( sin           µ cos       )
                                    dt     R

                                                                                                      :+          ,               /2
                                                        : '            ' !     f       V '! (1)              #,                 ,! /
                                         dv                           dv
                                                   g sin = 0             = g sin
                                         dt                           dt
                                                                   d          v
                                                                  : = =          5         , J d                            " ?!
                                                                   dt         R
                             d
                                dv = g sin .d
                             dt
                                                                 vdv = gR sin .d               ( 3)
                             d         v
                                = =
                             dt        R



A.FIZAZI                                            Univ-BECHAR                                                   LMD1/SM_ST
        Dynamique du point matériel                                                     181

                        : [0, v ]           v    T' #                        [0, ]                           T' #           5#        ( 3) #,                                 #!
                                  v
                                                                                       1 2
                                      vdv = gR sin .d                                    v 0 = Rg ( cos                          cos 0 )
                                  0                  0
                                                                                       2
                                                                                                                                                        :        B
                                    v 2 = 2 Rg (1 cos                        )          v = 2 Rg (1 cos                 )          ( 4)
    #T'                 5         E#%'!@ "                :-                 +                                    #T' #$ 5     ' 0                           & /"
                                                                                                                        . ,!' N ,;                      - (# +
                                                                                                                         : N &! ( 2 )                   #,      ,!
                                                                                       v2                                        v2
                                            N + mg cos = m                                         N = mg cos               m
                                                                                       R                                         R
                                                                                                                                                   :#$' &% v 2 I ,!
                                                                  2 Rg (1 cos                       )
                              N = mg cos                  m                                                   N = mg ( 3cos               2)
                                                                                   R
                                                                             :          +                             #T' #$ 5                 '                      #' #%
                                        mg ( 3cos         0           2) = 0                cos         0   = 2/3       0   = 48°
-             & )*!% @                      ' #% @ : ,'' @ - B E7 5 9'!' !     - #% <             : -
                                                 .  , v ( 0)  '%@          ' 5     . %7# > # '% @
          .       '%@                   v ( 0 ) D=     +            - #T !       v0 : v0 % v ( 0 ) :"
                                                : 5      # %      !     , O     '%@               !# 7 # 5
                                                                                        2 v (0)
                                                                                                              2

                                                                       cos         0   = +
                                                                                        3 3Rg
                      .m '                      &'             &%' #$!5                 O g                 R D v ( 0 ) C% : ,''           0                #        E7
                                                                                                                                  : % #!                        "#            /8
                            v = 2 Rg (1 cos
                            2
                                                                  )
                            0                                 0
                                                                                 v0 = 2 Rg (1 2 / 3) , v0 = 3, 65ms                            1

                            cos     0   = 2/3


              M                                             M                               v0 x                                                    f
                                                                                                                                          M0                     N
                                                                                                    X                                                   M
                            FN                                                     0                                                  R                     PT
                                                                      v0 y                    v0                                               PN                     +
    FN                                                    0

                                        T                                                               T                                                                 T
                                                                                                                                                        P
                                                N                                                                                     O
    N
                                                                      Y                                                                N
                  P
          ( )                                                         ( )                                                                      ()

         . ? B          %7#             &            ; 7(                         #5               ! .+      #$' #T !                                               /3
                                                                                                       .(" . ) MXY ,                                             M !/
                                        Fx = 0           vx = v0 .cos                   0           ( 5) : 1'!    &'                           :X C                :



A.FIZAZI                                                                  Univ-BECHAR                                                                   LMD1/SM_ST
   Dynamique du point matériel                                              182

                                                                                 : #1'!#% - T'                  &'                              :Y C                      :
                                                        Fy = P = mg                     a y = g = Cte
                                                   v y = gt + v0 sin            0        (6)
                                                                                         : ; 7&        1                                - . - #%                  F           ,!
               v =v +v
                2     2        2

                                                              v = g 2t 2 + 2 gv0 sin 0 .t + 2 Rg (1 cos                                             )
                      x        y

               v = 2 Rg (1 cos                     0)
                2                                                                                                                               0
                0

                                                                                                                   : $                                  >#,. - #% # 5
                                                                          v = v0 .cos 0 .i + ( gt + v0 sin                          0   )j
                                                                                        (8         . ):        1#!                      #                   ' & #' . /"
                                                           dv                             mg ( gt + v0 sin                 )
                              FT = m.aT = m                               FT =                                         0
                                                                                                                                            :                         .
                                                           dt                           g 2t 2 + 2 gv0 sin 0 .t + v0
                                                                                                                   2



                                                                                v2
  $        L#! !@        ( )*! B A 7                            FN = m                       !#&       # ,' #% +*! @ :                                  /             .
                                                                                r
                                                                                                                               . R !5 #&' @                               7
                                           P = FN + FT                FN = P 2                   FT2 - #%,                     - & E7 9'!' ! 7
                                                           g 2t 2 + 2 gv0 sin 0 .t + v0 sin 2
                                                                                       2
                               FN = mg 1                                                                       0

                                                                g 2t 2 + 2 gv0 sin 0 .t + v0
                                                                                           2




                                                                                                                                                                 11.50
       : &      I B                    %                )*'!                   W #*                     # !                ? B                    %7#             & -./
                         MT                                                    5,98.1024
             gT = G                        , gT = 6, 67.10                11
                                                                                                        gT = 1, 08.10 2 N .kg                            1

                                                                               (1,92.10 )
                               2                                                                   2
                          d                                                                  8

                          2
      : &       I B                %                 )*'!                      W #*                    # !                      &           %7#              & - . /"
                                                                                             22
                         ML                                                    7,36.10
             gL = G                        , g L = 6, 67.10               11
                                                                                                        g L = 1,33.10 4 N .kg                            1

                                                                               (1,92.10 )
                               2                                                                   2
                          d                                                                  8

                          2
   W #*               # !                      &        %7#           &                 ? B            %7#             &      9'#!                           &     - . /8
                                                                                                        : &                I B %                                   )*'!
                                           g R = gT           gL      ,        g R = 1, 07.10 2 N .kg              1


              : &         I B                      ' %7#             9'#!           &                  ,! =7 I B                                                    ,%        /
                                                                          ML                 MT              ML                     MT
                    gR = 0                 g L = gT , G                                 =G                                     =
                                                                     (d        r)                         (d       r)
                                                                                    2                                      2
                                                                                             r2                                     r2
                          r2                   MT               r2
                                           =                                = 81, 25
                    (d        r)                           (d      r)
                                   2                                    2
                                               ML
                      r
                               = 9, 01               r = 3, 45.108 m                     r = 345000km
                    d r



A.FIZAZI                                                        Univ-BECHAR                                                                             LMD1/SM_ST
   Dynamique du point matériel                                            183


                                                                                                                                            12.5
    #!                   #,'                                   2 &               #& % . A           &!            - /P        2 &           .         #! /
                                                                                                                                            :       ' #
                                                                 P0 = T cos
             O                 y'                                                                            l0
                                                                 P0 = kl0                           l1 =
                  T                                                                                        cos
                                                                 T = kl1
                           T cos
                                                                 F = T sin
         x                 A            x'
                                                                 F = kl2 = T sin
                 T sin              F                                                                         kl0
                                                                    P                        kl0                  sin              = kl2        l2 = l0tg
                           P0                                    T= 0                     =                  cos
                                                                   cos                      cos
                               y

                                                                                                                                            13.5
                                                                                                                    :                      - /P     - & /
                           F = ma = mr = 6 ( 6i                          24t. j ) , F = 36i 144t. j


                                                                                                                    : %            % ! #% - &             /"
                                                     i                   j              k
                           = r & F = 3t          2
                                                          6t         4t      3
                                                                                   3t + 2
                                                     36          144t                   0

                                (                         )
                           = 432t 2 + 288t i + (108t + 72 ) j +                                 (   288t 3 + 864t 2 k     )
                                                                                                                              :                             /8
                                         p = mv = ( 36t 36 ) i                              72t 2 . j + 18k
                                                                                                              : %             % ! #%                  ,
                                                 i               j                 k
                      L = r & p = 3t         2
                                                     6t         4t   3
                                                                                 3t + 2
                                         36t 36                 72t               18

                           (                             ) (
                      L = 144t 3 + 144t 2 i + 54t 2 + 72t + 72 j + 72t 4 + 288t 3 k             ) (                            )
                                                                                                                                       dp
                                                                                                                          :F       =      5           0'! /
                                                                                                                                       dt
                                                 p = ( 36t 36 ) i                      72t 2 . j + 18k
                                                 dp
                                                    = 36i 144t. j = F
                                                 dt
                                                                                                                                       dL
                                                                                                                              : =               5         0'!
                                                                                                                                       dt




A.FIZAZI                                                       Univ-BECHAR                                                                 LMD1/SM_ST
   Dynamique du point matériel                                        184

                         (                          ) (                     ) (
                    L = 144t 3 + 144t 2 i + 54t 2 + 72t + 72 j + 72t 4 + 288t 3 k                                            )
                        = ( 432t         2
                                             + 288t ) i + (108t + 72 ) j + ( 288t                    3
                                                                                                                    )
                                                                                                         + 864t 2 k =
                                                                                                                                             14.5
                                                                                               : R C % ! #% M                                       %,' /1
                                                       OM = r = lur
                                                       v = r = lur
                                                                                    v =l u
                                                       ur = u


                        : ( O, ur , u , u z ) - #&                    O              &!       % ! #% M         &!                            , "      ! /2
                  LO = OM & p
                                                                           ur             u     uz
                  p = mvr + mv
                                                             LO = r = l                0        0 ; LO = ml 2 .u z
                  vr = l.ur = 0 ( l = Cte )
                                                                    0                 ml        0
                  v = l .u

 % ! #% M    &!             &%               2 &             "#                      %@                    ,        1! : % ' #!                 ! '       '
                                                                                                     : ( O, ur , u , u z ) - #&                 O        &!

                   O   = OM & T + OM & P           (              )                     ur                   u                    uz
                                     0

                  P = Pr + P                                                    O   =    l                   0                    0
                  Pr = mg cos .ur                                                     mg cos               mg sin                 0

                  P = mg sin .u

                                                         O   = mgl sin .uz
                                                                                                                :                      1        / 23
                    dLO                                                                             g
                        =        0       ; ml 2 .u z = mgl sin .u z                             +     sin = 0                    (1)
                     dt                                                                             l


                                         : ( O, i , j , k ) - #&                O         &!     % ! #% M               &!                     , "            !
                                                             i              j        k
                       LO = OM & p
                                                        LO = x             y         0 ; LO = m ( xy                yx ) k
                       p = mvx + mv y
                                                             x             y         0
                   : ( O, i , j , k ) - #&                   O        &!            % ! #% M          &!                &%             2 &           "        !

                   O   = OM & T + OM & P         (               )                    i  j            k
                                 0
                                                                            O       = x y             0 ;      O    = mgx.k
                  P = Px + Py = mg . j                                                0 mg            0
                        0




A.FIZAZI                                                     Univ-BECHAR                                                                   LMD1/SM_ST
   Dynamique du point matériel                                                       185

                                                                                                                                          :                     ,       1! :% !
                 dLO
                     =      0    ; m ( xy + xy xy                              yx ) k = mgx.k                       xy           yx = gx                   ( 2)
                  dt
                                                                                                                                  : #' # '                   ' '!          5 :& '!
                           x = l sin                   ; x = l cos                            ; x = l cos                l        2
                                                                                                                                      sin
                           y = l cos                  ; y = l sin                             ; y = l sin                    l        2
                                                                                                                                          cos
                                                                       : (1)              #,           ! ( 2)            #,                                         *#!, I ,!
                                                                                                           g
  N                                                                                                    +     sin = 0                           ( 3)
                                                                                                           l
      O
                                                                                              :+              ,                                                         4 23
                                                               T                ' ' P #$ &/ : ' & 1           m                                                       ' J? '
                   T
                       M
                                                                                               . F C% # $' *                                                             !H %
                                                                             . ) 1#!       #     '%         *                                                            ' #!!
             u                  P cos
                                     u                                                                                                                                 :(:
                   P
                                                                                     F = P + T = ma
                                                                                                                                  P + T = FT + FN
                                                                                     F = FT + FN = ma
> # '              > # '                      % (<, A 7                                                                                                         % (<, ) ,!
                                                                                                                                                                     :=
                                                                             dv           v2
                                              v = l , aT =                      = l , aN = =                              2
                                                                                                                              l
                                                                             dt            l

 ' & #       . OZ                            % ! #% 2 &                                   #      #!!           Dm '                     !                             # 5 #!!5 # %
    ." #     $ #' #%                          #      &/                          .                              # (<'                 ' &         B #                , T        FN
                                         =        P
                                                      +    T
                                                               =        FT
                                                                             +       FN


                                         T
                                              =       FN
                                                           =0
                                                                                                 mgl sin = m l 2
                                         P
                                              = P.l sin

                                         FT
                                              = FT .l = m l 2
                                                                                                           :                              #,                * ! 7
                                                                         g
                                                                   +       sin = 0                     ( 4)
                                                                         l
                                 .                           5 6 7                              ( 4)          (1)            ,
                           :#!                             1#!                                        #& G#% . P + T = ma 1                                                #!   /3
                                         mg cos + T = maN                                      T = mg cos + m l                       2


+%*' ( sin ' )              - T* ,                                 7                  '           5           .1                                      T'  ' ' 5 1 <!
                                                                                                                     : .                              (1) ?#;' #,
                                                                                     g
                                                                                 +             =0
                                                                                     l
                                                                                                                                                            :          J #. #$



A.FIZAZI                                                               Univ-BECHAR                                                                                LMD1/SM_ST
   Dynamique du point matériel                                            186

                                                                                        g
                                                                  =       0   sin         t
                                                                                        l
                                                                                                                        :                                    !
                                                                              g     g
                                                              =   0             cos   t
                                                                              l     l
                                                                      :                            ,!'              ' J?                      M !                  !
                                                                  g                           g
                                     =0               sin           t =0                        t = 0 ± k(
                                                                  l                           l
                                                                                                                            : 1 5                          ' #$!
                                           g
                                     =       cos ( 0 ± k( )                                                         g
                                           l
                                             0
                                                                                                       =       0
                                                                                                                    l
                                 cos ( 0 ± k( ) = ±1
                                                                                                                   := #              # 1 5 ' ' +%*
                                                                                          g
                                                       T =m g+                      0
                                                                                          l
           '   5                 5 =5 D                       J &! @ ' T ' '                                            E           '"       .    7
                                                                                                                        .            ! @ ' - . E7 (B

                                                                                                                                                 15.5
E7 #!' #       .                         #!                                                        , >                 = #                                        ,
D ''     #              H       DG               &!               = #            (L ) O  O/G                        &! % ! #%                                     ,
                               . G C % ! #% ( LB / G                          ) B (L ) A           A/G                  ' &!                      OC        % ! #%
                                                  LO = LG / O + LA / G + LB / G
                                                                                                                                         : ( LO / G ) "#     % 5 %!
                              LG / O = OG & pG / O
                              pG / O = 2mvG / O
                                                                              LG / O = 2m OG & vG / O (                         )
                                         i                                          j                     k
                   LG / O =    xG = a cos             1               yG = a sin              1            0 = ( xG yG              xG yG )
                              xG = a 1 sin                1       yG = a 1 cos                    1        0

                                                              LG / O = 2ma 2                  1
                                                                                               2
                                                                                                            (1)
                                                                                                                        : ( LA / G = LB / G ) F "             !
                               LA / G = GA & p A / G
                               p A / G = mv A / G
                                                                                                   (
                                                                               LA / G = m GA & vA / G                       )




A.FIZAZI                                                  Univ-BECHAR                                                                           LMD1/SM_ST
      Dynamique du point matériel                                          187

                                          i                                             j                         k
                    LO / G =    x = d cos
                                    '
                                    A                    2             y = d sin
                                                                           '
                                                                           A                      2                    ('
                                                                                                                  0 = x A y 'A            x 'A y A
                                                                                                                                                 '
                                                                                                                                                     )
                               x = d
                                '
                                A              2   sin       2     y =d'
                                                                       A                2   cos           2       0

                                                                 LA / G = md 2                2
                                                                                              2   = LB / G                     ( 2)
                                          :"                                       * ! ( 2)                           (1) ' #%, J ! 5 @ #! &% #
                         LO = 2ma 2           1
                                               2
                                                   + 2md 2         2
                                                                   2           ,        LO = 2m a 2               (    1
                                                                                                                        2
                                                                                                                               + d2   2
                                                                                                                                      2   )
                                                                                                                                                               16.5
  #      ( )*! 1'!                                 %     &'                             ' '           #$ &/ ,?# M                               &!         1                  /1
- (     ! 9'! = &                                                  ' &                  #&            . AZ                                           OXY 2 '
                                                                                                                                                      . T sin
                                                                                                                                                                      :
                                        T + P = ma
                                        T sin           = ma N = m                  2
                                                                                        r
                                                                                                    T =m               2
                                                                                                                           l
                                        r = l sin
                                                                           : ; 5:                                     .               #(< ! #                  !              #5

                                                                                                              2
                                                       T sin                   sin                m           l sin
                                        tg =                                                =
                                                         mg                    cos                            mg
                                                                                                g
                                                                           cos          =         2
                                                                                                      l
             :O %         7 !              B            #/        G#% A C % ! #% M C                                                             , - #% "#                    /2
                                          LM / A = AM & p
                                          AM = AO + OM = zu z + rur
                                          AM = l cos .u z + l sin .ur
                                          v = zu z + zu z + rur + rur = r u
                                                                  0

                                          v = v = l sin .u
                                          p = ml sin .u
                           ur                  u                      uz
  LM / A = AM & v = l sin                  0                       l cos                      LM / A = ml 2 sin                        ( cos             .ur + sin .u z )
                        0               ml sin                        0


               : M C % ! #% A                      &%             2 &              = # '
                                                                                    *          % ! #% '&'.                                                                5   0'!
                                                                 M / A = AM & F :
                                                                                  A &! % ! #% 2 &        "                                                                !     %
                                                                                                                                                                   : F '()*+,




A.FIZAZI                                                     Univ-BECHAR                                                                                    LMD1/SM_ST
   Dynamique du point matériel                               188

                                        F = T + P = ma
                                        F = T sin            = ma N = m                  2
                                                                                             r
                                        F =m         2
                                                         l sin .ur
                                                                                                                              : AM >#,.
                                             AM = zu z + rur
                                             F =m             2
                                                                  l sin .u

                                             Z
                                             A
                                                              l
                                             uz                    T
                                                                        uz               Y
                                             O
                                                                                     u
                                                             T sin           M
                                        X                                                u
                                                                             P


                                   ur            u                 uz
            M/A   = AM & F =    l sin            0                l cos                  M/A         = ml 2     2
                                                                                                                    sin .u      (1)
                               m 2l sin          0                   0
                                                                                 :                    % ! #%                 , :#&'.#%      &!

                                 LM / A = ml 2 sin            ( cos        .ur + sin .u z )
                                 dLM / A
                                   dt
                                         = ml 2 sin                ( cos      .ur + 0            )
                                                                    :             * ! I ,' #%                          ur = .ur : 5 ) ,!
                                  dLM / A
                                          = ml 2         2
                                                             sin .cos .u                             ( 2)
                                    dt
                                                         :                    ,                  1!         *         #! 0' (         ! 7
                                                 dLM / A
                                                         =              M/A
                                                   dt
                                                                                                                              :17.5
    -7%#!     5- #      - &     B #          !                      '         +%*' =                      ) ,!                    # & /1
                                                                                                            .                ! M ! "7 '
  :!         ! 2 & E7            '. #         % ! #% M !                             - /P               2 & #! %             %#&    . /2
                                P + Fc + T = 0               P + Fc = T
                                                   v2
                                                              m
                                        Fc                                                         v2
                                tg =          tg = R                              R=
                                        P         mg                                             g.tg




A.FIZAZI                                     Univ-BECHAR                                                                      LMD1/SM_ST
   Dynamique du point matériel                                 189

                                                                2
                                                 120.103
        O                                          3600
                 T                           R=                                R = 631N                                                  :=      :% '
                                                9.8 × 0,176
                                         "                     *               #               # ( !#/                 /</ <              # & J & /3
                 R
                               Fc
                                                                                                                                              .#$%#
                                                         d = vt , d = 1000m : 30s                                       <                  &   #
                           P                                                                    :             %       '  # &             5 !, 7
                                                                                       d
                                                       d=R                         =     ,               ' 1,59rad ,            ' 91°
                                                                                       R
                                                                                                                                   :5.18
             B
                                         . P2        AB L                &/            P1                BC L                 &/     t 1
                                                                                        :                         A         '    # B 5 % :% !
    A                                                                   P + P2 = M 1 + M 2 a
                           x                                             1
                                                                                                          M

                                                !     ; B           !                           (#.                               #,. - #%,        & !
                                                                                                   : BC %                   L                    x C%
             C                                  P P2 = Ma
                                                 1

                                                M = +L                                                                                     dv
                                                                                                        + xg + ( L x ) g = + L
                                                P = M 1 g = + xg
                                                 1                                                                                         dt
                                                P2 = M 2 g = + ( L x ) g
                                    : !#/ )            7 !#/                                             ?#;'         #, #!              *' +          ' !
                                                dv
                               2 gx gL = L
                                                dt                                                        2g
                                                        Lx = 2 gx gL                                x        x= g
                                    dv                                                                     L
                               a=      =x
                                    dt
                                                                                                                  2                  g
                       :       '&         & #% x ?#;'           #,                     I ,! x = L                            a=               :& '!
                                                                                                                  3                  3
                                      2g
                                    x    x= g
                                       L                                   4g                                     g
                                                             x=a=                           g            a=
                                      2                                     3                                     3
                                    x= L
                                      3

                                                                                           :              #% & ,'               '!              F H %!
                     2g
            r2          = 0:            !#/ )          % !#/                                            ?#;'           #,       E7$ -                 #,
                      L
                                                                                                         2g                       2g
                                                                                       r1 = +                     ; r1 =             :#         # <
                                                                                                          L                        L
                                                                                       :                 ?#;'          #,       E7
                                                      2g                2g
                                                         t                 t       L
                                             x = Ae    L
                                                             + Be        L
                                                                               +                    (1)
                                                                                   2



A.FIZAZI                                              Univ-BECHAR                                                                        LMD1/SM_ST
   Dynamique du point matériel                                             190

            x=b
   t=0             #            7                   '%@                    .                 '!' ! # 7                      .B           A   '%#/        ' &%
            v= x=0
                                                                                 2g                              2g
                                                          2g                        t          2g                   t
                                                    v=x=A    e                    L
                                                                                             B    e               L
                                                                                                                                  ( 2) :                 - #%
                                                           L                                    L
                                                                                 :B          A               ! ( 2 ) (1) ' #,                            I ,!
                                                    L
                               b = A+ B +
                                                    2                                                            b          L
                                                                                              A=B=
                                            2g                  2g                                               2          4
                               0= A                     B                  A=B
                                             L                   L
                                                                                                                  2g
        *#            #(<,          ) #,'               #//                      ) ,! #                          =   J?! #%#                              $ '
                                                                                                                   L
                                                                       : 0 ( ch ) =                           #' "    ( sh ) =                           " #%
                                e   t
                                         e      t
                                                                           e t +e        t
                      sh t =                            ; ch t =                                 ; ch 2 t sh 2 t = 1
                                        2                                      2
                                                                                 : #'                !               ( 2)           (1) ' #, "' !
                               2b L e t + e                 t
                                                                      L               2b L        L
                      x = 2.                                      +             x=         ch t +                                 ( 3)
                                 4      2                             2                 2         2
                                                            t          t
                                 2b L                   e        e                               2b L
                      v = x = 2.                                                 v=x=                 sh t                           ( 4)
                                   4                            2                                  2
                                                                                                                                                    2
   : ( 3)     #,          =                 #' "                - #% 8           ' !             ( 3) #,                          I ,! x = L              5#%
                                                                                                                                                    3
                               2     2b L        L                                                       L
                                 L=2      ch t +                                 ch t =                                     (5)
                               3       4         2                                               6b 3L

                                                                                         :=                  "              8       ' ! ( 4)        #,
                                    2b L                                                     2v
                       v=x=              sh t                         sh t =                                                       (6)
                                      2                                              (   4b + L2 4bL
                                                                                             2
                                                                                                                        )
                   :# $, % ' ,% )                       )             ( 6 ) ( 5) ' #, J ! . ch 2                                    t sh 2 t = 1 5 #!
                                                        2
                                            L
                      ch 2 t =
                                    6b 3L
                                                                            2
                                                2v                                                                              2 2
                       sh 2 t =                                                   v2 =           2
                                                                                                             b 2 + bL             L
                                        (   4b + L2 4bL
                                              2
                                                                       )                                                        9

                      ch 2 t sh 2 t = 1




               :#$!       0' #'             #       '            &              )#               #$!                 * ! #$' &%                 I ,!        ,!



A.FIZAZI                                                        Univ-BECHAR                                                                   LMD1/SM_ST
         Dynamique du point matériel                                           191

                                                                   2g                                 2 2
                                                          v=                   b 2 + bL                 L
                                                                    L                                 9
                                                                                                              b = 7m        L = 12m := , : % '
                                                                       v ' 10, 6ms               1


                                                                                                                                      :               . 3
                                                                                                                     2g
     : x *#;               @%                  - #%               * '!                  #'            ,%      Dx        x= g          #,          #(< !
                                                                                                                      L
                    2g                     dv 2 g                          dv      2g
              x        x= g                   =   x g                         dx =    x g dx
                     L                     dt   L                          dt       L
              dx      2g                                               2g
                 dv =    x g dx                           v.dv =          x g dx
              dt       L                                                L
              v            x
                                   2g                      1 2 g 2                                            g 2       g
                  v.dv =              x g dx                 v = x                      gx           v2 = 2     x 2 gx 2 b 2 + 2 gb
              0            b
                                    L                      2    L                                             L         L

          2
    & . L = x <                             +%            9 .                               :                  "%; 2 3           2.               4     .
          3
                                                                                                                            :         !* ,             >

                                                                   2g                                 2 2
                                                          v=                   b 2 + bL                 L
                                                                    L                                 9


                                                                                                                                           19.5
    (OM , Oz ) =                   1                                                             +                  M       &! J?           # # $ /1
                                                                  r r0                               r0
                                                      tg =         =                    z=r                                                       : #' #%
                                                                  z z0                               z0
                                           :          !           B        #/           G              M      &! > # ' 5 M                      ) ,! /2

                                                a= r r             2
                                                                        ur + r + 2 r                        u + zu
                                                             ar                              a                 az

                                                                                        r0
#             #         &!                 #' /P            #' & . z = r                   :                            +                  &!         &% 7
                                                                                        z0
    '%                7 R              +              ,                - (               D P = mgu z -           %       =7 P #$ &/
                                                                                               . R = Rr + Rz = R cos .ur + R sin .u z
    :             * !          !       @       ,          /</          #                     ' & & ! /A '           # B 5 % :% !
                           P + R = ma = F
                           F = Fr + F + Fz = m r r           (             2
                                                                               )u   r            (
                                                                                        + m 2r + r             )u    + mzu z    (1)
                           F = R cos .ur + R sin .uz                           mguz
                           F = R cos .ur + ( R sin                         mg ) u z                  ( 2)



A.FIZAZI                                                          Univ-BECHAR                                                             LMD1/SM_ST
      Dynamique du point matériel                                            192

                                    : #'          /</             ?#;'               @ #,                              * ! ( 2)               (1) ' #,              &%# %

                                                        R cos          =m r r    (               2
                                                                                                     ) ( 3)
                                                                 (
                                                      0 = m 2r + r                       ) ( 4)
                                                                                             z0
                                                       mg + R sin                    =m         r              ( 5)
                                                                                             r0

                                                                                 z
                                                                             H                    uz
                                                                                                                   u
                                                                                                     M0
                                                                                                               u
                                                                                                  z0
                                                                             O
                                                                                             0                          y
                                                                                     0
                                                          x

  7         .               % ! #% r 2       &        :'.                        2r + r = 0                            &            5 9'!' ! ( 4 ) #,       /3
                                                                                                                                     :r = C 5
                                                                                                                                       2       te
                                                                                                                                                      #!% = P
                                             ( r ) = 2rr.
                                                          '
                                                  2
                                                                           + r2 = 0                      r 2 = C te
        :       '%G                 - #% 9'!' ! #$!                          !            B          #/            G                1                     - #% ) ,!
            v ( t ) = r ( t ) ur + r ( t )       (t ) u       + z ( t ) ur           v ( 0 ) = r ( 0 ) ur + r ( 0 )                     (0) u   + z ( 0 ) ur
                                                                                                                               : 7                '%G                 -.
                                         v (0) =              r ( 0)       + r ( 0)              ( 0)        + z ( 0)
                                                                       2                                 2                      2



  .                    '%         .% @                  O 7 . z ( 0) @ r ( 0)   % - #% #! I ; V!
            :               '%@                       7$ #!' . 0      (#%     ? ';! # 7 . r (0) = z ( 0) = 0
                                                                  v (0) = r ( 0)                 (0)
                                                                                                       :H %                    '          *               7           #,%'
                                                                           (t ) = r (0) ( 0)
                                                                       r (t )
                                                                                 2                             2
                                             r 2 = C te
                                                       r ( 0 ) .r ( 0 ) . ( 0 )
                                               (t ) =
                                                                r (t )
                                                                       2

                                                                                        rv
                                             v (0) = r ( 0). ( 0)                     = 0 20
                                                                                        r
                                             v ( 0 ) = v0 , r ( 0 ) = r0


  ( 3)          ' #,         M       D r (t )                   J%#' #$                  B           P             7                 !,            ( 5)        #,      /4
                                            : %#'                      ( 3) #,                           .#,            (t )         r (t )             ' '          ( 4)




A.FIZAZI                                                          Univ-BECHAR                                                                        LMD1/SM_ST
   Dynamique du point matériel                                        193

                                                           1         z0
                                                R=           ! mg + m r "
                                                         sin         r0
                                                   :              * ! ( 3)                   #,           -          B - #%, E7$% R I ,!
                                               2
                                             v0 r04 1                        z0 r0
                                  r                  . =                             g             ( 6)
                                           r02 + z0 r 3
                                                   2
                                                                            r + z0
                                                                             0
                                                                              2    2


                                          A( r0 ,v0 , z0 )               A( r0 , z0 , g )

 5 #    D r (t ) = r ( 0)     1                               5         !,         7$             1'!                                   !#     7 /5
                                                                                             :                '' ( 4 ) - #%, . ( ( t ) = C te )
                              v12 r04 1      zr                                       v12      z
                                       . 3 = 2002 g                                         = 2 0 2g
                            r0 + z0 r0 r0 + z0
                              2      2
                                                                                  r0 + z0 r0 + z0
                                                                                    2     2


                                                                                             v1 = 2 gz0 : 7                        %

                                           2r
                            2rr + A           = 2 Br :#!                    +%* 2 C% ( 6 )                         #,      " ?           ,%     &! /6
                                           r3
                               2r                                                 A
                 2rrdr + A        dr = 2 Brdr                          r2            = 2 Br + C                ( 7 ) : ! 9'!                 B #$ # '
                               r3                                                 r2
           : t = 0, r ( 0 ) = 0 E< 5 #$                      #.                 '%@               .           J          ! C %#/                *
                                          A                                                 A
                                  0          = 2 Br + C                      C=                    2 Br
                                          r2                                                r2
                                                                                                              :+%*' ( 7 )          #,           B
                                                             1          1
                                      r2 = 2A                              + 2 B ( r r0 )
                                                             r2        r02


                                                                                                                                         :20
           :> # '                 DJ?                    >#,.           #%                        %,'     ' !              '     ,' !
                                                   x                    x                    x
                                            r y              , v y                 , a y
                                              z                  z                     z
                                                                                                                               :- & - #% "          !
                                            i            j    k             0
                            v&B= x                     y          z = Bz
                                            B          0          0             By

                                      (                      ) (
                             F = q E + v & B = q Ek + 0i + Bzj                                          Byk    )
                             F = q 0i + Bzj + ( E By ) k                                          (1)
                                                                             :"' ! 5                      A          '         # B (<, : % '%
                            F = Fx + Fy + Fz                          F = mx + my + mz                        ( 2)



A.FIZAZI                                               Univ-BECHAR                                                                     LMD1/SM_ST
   Dynamique du point matériel                           194

                                              : ?#;'    @ #, H</                       #! 9'!' ( 2 )             (1) ' #,          &%#
                                                  mx = 0
                                                  my = qBz
                                                  mz = q ( E + By )
                                                                       : #'                 '%@              .     #%' @       ,% 7 0!
                     t = 0:
                     x ( 0) = 0 , y ( 0) = 0 , z ( 0) = 0 , x ( 0) = 0 , y ( 0) = 0 , z ( 0) = 0
                                                         : ?#;'          @ #,                 ! '            -                 "' !

                                  mx = 0        x=0      x = C te = x ( 0 ) = 0               ( 3)
                                                       q     q
                                  my = qBz        y=     Bz    Bz ( 4 )
                                                                    y=
                                                       m     m
                                                      q      q
                                  mz = q ( E By ) z = E        By ( 5 )
                                                      m     m
                                           : ( 4 ) #,   #$' &% y I ,!                                ( 5)        ?#;'       #,

                                               x=0     (6)
                                               y= z      (7)
                                                         2
                                                    q             q
                                               z+ B          z=     E         ( 8)
                                                    m             m
                                                                                                                             q
                                                                   :      (8 ) ?#;'                     #,              =B     J?!
                                                                                                                             m
                                                                                        2
                                                                 qE m
                                        z=     sin t + , cos t +
                                                                 m qB
                                                                       mE
                                        z=     sin t + , cos t +                        (9)
                                                                       qB 2
                                       : ' #,        # ,' #%        '%@                .          #(< ! ,                   '%#/         ,!
                                                      mE
                               z=       sin t + , cos t +
                                                     qB 2
                               z=        cos t , sin t
                                                                                              mE
                              t = 0 , z ( 0) = 0 , z (0) = 0            =0 , , =
                                                                                              qB 2
                                                                                       : z ( t ) - #%              * '!            B
                                        mE          mE                       mE
                            z (t ) =       2
                                             cos t + 2            z (t ) =        1 cos t
                                        qB          qB                       qB 2
                                                                               a

                                                       z ( t ) = a (1 cos          )
           : y ( t ) - #%              * '!     # ! / D z I ,! ( 7 )               #,             . y (t )        #,         ' #!        &%




A.FIZAZI                                          Univ-BECHAR                                                          LMD1/SM_ST
   Dynamique du point matériel                  195

                          y = a (1 cos     )        y= a       a cos t

                          y ( t ) = a ( t sin t )      y (t ) = a (      sin   )
                                                                                          : #$!
                                        x (t ) = 0
                                        y (t ) = a (    sin    )
                                        z ( t ) = a (1 cos    )
                                               .0      , #         - X             @ #,           E7
                             z
                            2a




                             O                                        2( a x




A.FIZAZI                                 Univ-BECHAR                                  LMD1/SM_ST

				
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