# dynamique

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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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