Correction Des Exercices Systeme Oscillant Classe 1

8/8/2019 Correction Des Exercices Systeme Oscillant Classe 1

1/12

Allal mahdade http://sciencephysique.ifrance.com

Page 1

1

1 1 0FP

====++++ Oz0Kmg ====

K

mg==== cm5,2====

2 1

Oz amFP

====++++ OzzmKmg ==== ====++++tz

0Kmg ==== zmKz ====

0z

m

Kz ====++++

m

K20 ==== 0zz

2

0 ====++++

:

m

K0 ====

2 2 )tcos(Zz 0m ====++++

m10.4Z 2m

s/rad200==== ====

t=0z=Zm 01cos ========

)t20cos(10.4z 2====

2 3 m

KZV m1 ====

t z(t))tsin(Zv 00m ++++====

0m1 ZV ====

k

km

K

ZV m1

m ====

K

ZV m1 ====

V1=0,8m/s3

a=gz0=0 V0=-V1

t8,0t5z 2 ++++====

2

1 ( )S

( )S


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Page 2

( )S .

( )S:

P

R

( )S.

( ) ( )( ) ( )

.

0 .

M M

M M

P R J

R P J

+ =

= =

( ) 2 .

sin

M P mg OH

OH OG

=

=

: ( )S OG

1 2

2

mOG mOGOG

m

+=

1

G 2

G

1 . O 5OG Ri=

2 11OG Ri=

168

2

ROG i Ri= =

=8OG R

( ) ( )2 . sin 16 sinM MP mg OG P mgR = =

:16

16 sin 0 sin 0mgR

mgR J J

+ = + =

10m

sin =16

0mgR

J

+ =

( )0

2cosmt t

T

= +

.

:0

2 116

JT s

mgR = =

2 ( )S:

( ) 0

2

cosmt tT

= +

10 18m rad

= =

0t=( )0 cos cos 1 0m m = = = =

( ) ( )cos 218

t t

=

5 t

( )2 2 2 201 1

sin 22 2

C C m E J E J = =

EC


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Page 3

( )2

2 2 2 2 2

0 0

2 2

0

1 sin(2 ) 1 sin 2 1

1 1sin (2 )

2 2

1

2

m m

C m

t t

J t J

E J

:2 20

1

2C mE J =

:ECmax=6.10-3J6 S:

2 2max 0

1

2C mE J =

2 2 0

1

2

m mE J =. t

2 2 20 0

1sin ( )

2m m p E J t E = +

2 2 2 2 20 0 0

1 1sin ( )

2 2m p m J t E J + =

( )( )

( )

2 2 2

0 0

2

max

11 sin

2

cos 2

p m

p C

E J t

E E t

=

=

3

I 1 RP

:

( ) ( )M MP R J

+ =

sin2

sin 02

L Mg J

MgL

J

=

+ =

2 )

(

10

m rad

sin =

:

02

MgL

J

+ = 21

3 J ML

=

30

2

g

L + =


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Page 4

: 3

0

22

3

LT

g= :0 1,26T s=

II 1

:p E Mgz C = +

0pE =0z 0C= = p E Mgz=

( )1 cos2

Lz =

( )1 cos

2p

MgLE

=

2 : . 0====J5,0EE Cm ========

0Ep

====

s/rad77,5J

5,025,0J

2

1m

2

m====

========

J25,0E C

====

0, 5 0, 25 0, 25m p C P m C E E E E E E J = + = = = rad3

====

: 0EEE

cpm>>>> t .

CE maxCE

0 =min

0pE =

max

2

max max

21

2

mm C

E E E J

J

= = =

2 2 21

3.10 .3

J ML kg m

, :1 mE= = 5mE J=

max

210,8 / m

Erad s

J

= =

minCE

min max min maxm c p C m p E E E E E E = + =

( )

min

max2

max min

21

2

m p

m p

E E J E E

J

= =


5/12


Page 5

:

1,75m

E J=max

1, 5pE J=:( )max

min

24, 08 /

m pE E

rad sJ

= =

4

P 0

( )S 1

( ),O k

.

1

P

F

G

0P F+ =+ =+ =+ =

Oz

0 0eqP kG G =

1 2

1 2

( ) 0

( )

1

m m g k

m m g

K

cm

+ + = + + = + + = + + =

++++ = = = =

= = = =

2 1 :

1 2( ) F P m m a+ = ++ = ++ = ++ = +

( ) ( ) ( )

( )

0 1 2 0 1 2

0 1 2

eq eq

eq eq

kG G P m m a k G G G G P m m a

kG G kG G P m m a

+ = + + + = +

+ = +

0 0eqP kG G =

( )1 2eqkG G m m a = +

Oz:

1 2( ) kz m m z = + = + = + = +

:1 2

0k

z zm m

+ =+ =+ =+ =++++

:1 20( )

2m m

Tk

++++

====

:0T

0 0, 2T s=

: 22


6/12


Page 6

G :

t=0(0) cos 0, 2 cos

m m z Z Z = =

t=00

0

0

2( ) sin( )

20 (0) sin

m

m

z t Z tT

t z ZT

= += += += +

= = = = = = = =

((0 0z


7/12


Page 7

1 2

1 2

1 2

( )0 ( ) 0

( )

0( )

m dE t

m m z kzdt

kz z

m m

kz z

m m

==== + + =+ + =+ + =+ + =

= = = = ++++

+ =+ =+ =+ =++++

. O V23z=0= Zmz

2 2

1 2

1 10 ( ) 0

2 2m

kZ m m V + = + ++ = + ++ = + ++ = + +

2

1 2

mkZ

Vm m

====

++++

V=1,2m/s

zi>z :

S 1 2

( )P R m m a+ = ++ = ++ = ++ = +

Oz1 2 1 2

( ) ( )m m g R m m z + + = + + + = + + + = + + + = +

1 2

kz z

m m= = = =

++++

z

1 2 1 21 2

1 2

( ) ( )

( )

km m g R m m

m m

R m m g kz

+ + = + + + = + + + = + + + = +

++++

= + = + = + = +

0R 1 2


8/12


Page 8

2max 2 max1

2m

E kZ m gZ = += += += +

2 2

max 2 max 22 0m m E E Z m gZ m V ==== + =+ =+ =+ =

2 4 0, 29 0m mZ Z+ =+ =+ =+ =

max 0,07Z m====

5 1 :

:

P

R

.

( ) ( )M M MCP R J + + =

( ) 0M P =

( ) 0M R =

MC J C J = =

:0C

J

+ =

.

:0 2J

TC

=

2 0 0,92T s=:2

2 2 2 1

0 2

0

44 0, 233.10 . .

J JT C N m rad

C T

= = =

3 : :

:

P

R

)1( )2. (

( ) ( )21

M M M M C C

P R J

+ + + =

( ) 0M P =

( ) 0M R =

21

M MC C

J

+ =

1 1MC C= 2 2.MC C = .

1C 1K

CL z

=

2

KC

z =

L :K

CL

=

( )1 2 1 2C C J C C J = + = 1C2C 1 1

K J L z z

+ =

:.K C L=

( )

( )

2

2

1 1.

.0

.

LC L J C J

L z z z L z

L C

z L z J

+ = =

+ =

: 0T


9/12


Page 9

( )( )00 2

.2

.

z L z J TT z L z

L C L

= =

0T 3

Lz =

0 00

2 2 0, 433 3 3

T TL LT sL

= = =

0T maxz z=

: 0T

( )

( )

002

0 2 02

T L zdTL z

dt L z L z

= = =

0

T 2

Lz =

0 0max

0,462

TT s = =

6 1 :

CteC2

1E 2p ++++====

0E 0==== p 0Cte ==== ====

2

p C2

1E ====

.

:)equilibre(E)0(E mm====

)0(E)0(E)0(E Cpm ++++====

0)0(EC ====2

mp C2

1)0(E ====

2

mm C2

1E ====)1(

0E p max====2

C J2

1E

==== 2

maxm J2

1E

====)2(

)1()2(

J

CJ

2

1C

2

1mmaxmax

22

m ========

:s/rad66,1max ====

2 t


10/12


Page 10

8

C3)t(E

8

CC

2

1)t(E)t(EC

2

1J

2

1

2

m

C

2

m2

mCC

22

max

====

====++++====

:8

CEC

2

1E

2

m

p

2

p

========

:J10.042,0E 4C

J10.014,0E==== 4p

====

7

1 A: ====

J)F( iMMMM

A:P

A T R .

J)T()R()P( ====++++++++ MMMMMMMMMMMM

RT

.

J)P( ====MMMM

mgd)P( ====

MMMM sinld ====

Jsinmg ====

A

0sinJ

mg====++++

: sin

:

0g

0m

mg2

====++++====++++

A .

0

T :0

2Tg

=

:T 0

0, 4 1, 26T s s= =

2 :

0J2

1J

2

1E

)terre(E)A(E)tige(EE

22

C

CCCC

++++++++====

++++++++====

A :)A(EE CC ====


11/12


Page 11

:22C2

C m2

1EJ

2

1E

========

: :ctemgzE p Oz====++++

0cte0E0z p ============

(((( ))))cos1mgE p ==== 2

1cos2

====

:

2mgE

2

p

====

:

pcm EEE ++++====

0

0 0

2cos

2 2sin

m

m

tT

tT T

= +

= +

:

2 2 2

2

2 2 2 2 2

0 0 0

1 1

2 2

2 2 1 2sin cos2

1

2

m

m m

E ml mgl

ml t mgl t T T T

= +

= + + +

l

g20 ====

2 2 2 2 2

0 0

2 2 2

0 0

2

2 1 2sin cos

2

2 2sin ( ) cos

1

2

1

2

1

2

m m m

m

m m

g E ml t mgl t

l T T

ml t t T T

E ml

= + + +

= + + +

=

CteEm ====

3 1 m


12/12


Page 12

(((( ))))

gl2

v

1cos

cosmgl2mgl2mv

cos1mglmv2

1

2

A

m

m

2

A

m

2

A

====

====

====

3

m

====

A :

: ====m

1cos m

==== gl2vA ====

s/m4vA

====

.

Correction Des Exercices Systeme Oscillant Classe 1

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