Mecpoint_1

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:!(LMD)

:!" "

!#$

%&'(

20082009

)!)))!))))$))))LMD / PHYSI UE-1

*../-&.1

:

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A. F

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AZ I

A.FIZAZI C.U.Bchar LMD/ PHYS1/ SM_ST

ii

!"#$%&'()'*+.

-"((.:[email protected] /01(2/

3% 4 536.

0((((789(. (:&;-"7 $%(

"; ( '= / $%/2 + > :

/ &3(( '2 % 4/ " "; +

29 ?(!.

"5"PHYS-1LMD8 "A4:

" :"

):""(: 4 5(."

#$%& % ' ($ ')* +$ % ,-./ #$&.

$(-= 37B(+'2(:3(-"

1". ( ( (" (: &;-" #3( $%

C.

-=3(+ :*;//;"

6/DE:. F.

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iii

/01(!0(0

*(2 3&1:3*4(50("6"Par o donc faut-il commencer ?

Si tu consens, je te dirai que tu dois dabord comprendre les mots.

(Epictte. Entretien)

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iv

6789:;

viii

.............................................................................ix ......................................x ...............................................................................xi .............................................................................1 I.8...................................................................1 A-I.%*'................................................................

1 1.......................................................................1 .!"........................................................1 #.!".......................................................1 $.%!".........................................................1 .&'".............................................................1 (.) !*+&....................................................2 2.9334............................................................2 ......................................................................2 #., (-./....................................................2 $.", , ,......................................................2 0%1.1.....................................................................3

0%2.1.....................................................................3 0%3.1.....................................................................3 0%4.1.....................................................................4 .1.....................................................................4 0%5.1.....................................................................5 B-I.346'...............................................................

5 1./*..........................................................

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v

5 2.12*.............................................................6 3.!3!4.......................................................7 0%6.1.....................................................................8 0%7.1....................................................................9 ."6'........................................................ II9 1............................................................9 2.+..........................................................9 3.520%.................................................

9 4."5.............................................................9 5.62...................................................12 6.5!..........................................................12 0%1.2.....................................................................12 0%2.2.....................................................................14 0%3.2.....................................................................14 0%4.2.....................................................................

14 7.)............................................................15 0%5.2......................................................................16 8.+)..........................................................17 0%6.2......................................................................17

9.+7 $78...............................................18 0%7.2......................................................................18 0%8.2......................................................................19 0%9.2......................................................................21 .:!9#';3................................................... III21 1.91...........................................................

21 2.91.:.....................................................

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vi

22 3.!%";....................................................23 4.9!%";........................................................24 5.9!%";....................................................25 6.!%";.......................................................27 7."!%";.......................................................28 .. ."3'................................................................ IV28 A.'3.............................................................28 1.8*..................................................................

28 2.2......................................................................28 3.

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vii

43 2.!15 + 38..........44 0%8.4......................................................................46 D.-85&3'........................................................46 1.9!%";".....................................48 2.!%";"........................................50 0%9.4......................................................................51 0%10.4......................................................................53 E.!'..............................................................

53 1.=............................................................53 2.8"+.................................................54 0%11.4.....................................................................55 3. !"9>.......................................................59 0%12.4......................................................................59 0%13.4......................................................................60 4.""......................................................65 0%14.4.....................................................................67 3'$9...................................................................82

!#-3':$ *2$!'.........................

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viii

PHYSIQUE GENERALE: Alonso-Finn ; InterEdition, Paris.1979

MECANIQUE: Hubert Gi et J.P Sarmant ; Tec Doc Lavoisier, Paris.

1995

COURS DE MECANIQUE: Jozsef Hering ; O.P.U, Alger.

INTRODUCTION A LA MECANIQUE: J.L.CAUBARRERE et autres ; O.P.U, Alger.

EXERCICES ET PROBLEMES DE MECANIQUE CLASSIQUE :LAGOUTINE A.et SEBAA M. O.P.U, Alger.

MECANIQUE CLASSIQUE : L.MARLEAU, Universit Laval, Qubec.Canada.2006

COURS DE DYNAMIQUE DES CORPS SOLIDES : M.DEVEL ,Universit de France-Comt.2003

OUVRAGES 0111111

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ix

cotetitre23Auteur4Edition26501511(6)''*#57

35502505'/86)9#5702501511:);)6)9#57205015058%:))585#57395025056)'*))*=(#5762502505Mcanique newtonienne

du point.

J.P Meullenet

B. SpenlehauerEllipses

53502505Mcanique du pointmatriel.

E. ElbazEllipses

37502505?@86)9#57

26502511(6)'*'#570.5005058%#$&:A$#57

41502505:7%6)A)$#57/1404938%5 6)

CD'*'E#4(G

145/493C,* 6):%*:DH4 H)5&

0I4:)):JCJQuelques ouvrages disponibles la bibliothque du C.U.Bchar

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x

2%K8J)L)physique()=(phusis)::D8%):GM.

8% ' ) @N C #) OJD .P%#@N QMDR')6M:J :JC,*:

S*/D23"?.

-55: 6)(mcanique)C$(chaleur)#(acoustique)#&(optique)G)(lectromagntisme).

-2)64TM4' 1..(:5(PHYS-1):6).

/G(PHYS-2): 8)G.

!(Mcanique classique):($!' '.

!!(Mcanique cleste):"!$D'9.

-5(lacoustique physique):@

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xi

))!))))G))!)(PHYS-1)Programme de mcanique

)30+B 130(

COURS: Ahmed FIZAZI E-mail :[email protected] ou [email protected]

Programme de mcanique ( 2 cours + 1TD) / semaine. VHG=67.5 heures. 6 crdits

I/ RAPPEL MATHEMATIQUE (2 semaines) :/I8)0"((

Les quations aux dimensions 9334Calcul derreurs

3>6'Les vecteurs 9

II/ CINEMATIQUE DU POINT (3 semaines) : (RSTUV3 )WXYTZ:;W[\:;]T^7_ /IIMouvement rectiligne 'Mouvement dans lespace -85&'Etude de mouvements particuliers

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Analyse dimensionnelle bcd:;efgh:;


1

1//: #$)7 #$17

:'O)7J:

)'C47)

C$

)C

C47

MLTIiNJ

LC$

kilogramme

-)mtre

seconde

@ampre

=degr kelvin

%)mole

'candela

E

C$

kgmsAKmolCd

H/(3 :(units drives)#$7% ')) Q;C)M()M/#$ +4.

1):)N(')J(/)H.............(U/) :(units secondaires)#$ / #$ H L

@I.1):)l(7)C(C$)cal.........(

/'= 6:(unit supplmentaire):D C$)rad.(

#/ 2"':(multiples et sous multiples)m;nop;:

IJKLMNO10-110-210-310-610-910-1210-1510-18

qUp;prfixedcicentimillimicronanopicofemtoatto

PJQNOdcmnpfa

A-I/(111'111$11ANALYSE DIMENSIONNELLE

I.$('RAPPELS MATHEMATIQUES

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2

2"':

'10+110+210+310+610+910+1210+1510+18

prfixe.dcahectokilomgagigatraptaexa

dahkMGTPE

2/$::/I(:

$.:J:6)$(quation aux dimensions)G(grandeur)')4): M$$:

)1.1(][G M L T

=

?$T, L , ML::)(masse)'(longueur)(temps).

A/6#$%6DKL'&:DCQMDC7%CC$)unit(G:J

#$(Systme International des Units : S.I):):

)2.1(kg m s

U/Ir,s,tJ/$$U,V,W*(G.

MDW2#E;L#&C')2;L6E2.

)+:

)1.1:/#M(vitesse)T(acclration).

:[ ] [ ] 1x LV V V LT t T

= = =C$:1ms

T:[ ] [ ]1

2. .v L T

a a a L T t T

= = =C$:2. sm

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3

).12:C/#M(force)'(travail).

[ ] [ ] [ ] [ ] 2.F ma F m a F MLT= = =C$:2kms=[ ] [ ]2 2 2. .W F l W MLT L W ML T = = =C$:2 2 .kgm s N m J = =

).13:%@)/#M(capacit dun condensateur).

6)L*$QMD:J.Q;C)MCH.#M CE '* G) :J4:/ JX 6M

C4(intensit) OL(M1I.

/#MY&:

)3.1(][ G M L T I =

A:

[ ] [ ]

[ ]

[ ][ ]

[ ] [ ]

[ ]

2 2

2 2

.

QQC C

V V

Q It Q I T W Q.V W ML T

W ML T V

Q IT

= =

= =

= =

= =

[ ] [ ] 1 2 4 22 2IT

C C M L T I ML T

IT

= =C$:Z1 2 4 2 ( )kg m s A F farad =

).14:/#M$(permittivit)Y%@).)'DH221 + CmNK

A:

2:[ ] [ ] 1== LCd

SC

:[ ] 2 1 2 4C I M L T =XJO:[ ] 2 1 3 4I M L T =

C'$L38:

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4

[ ] ( )( )( )2 2 1 1 2 2I T M L T L =

[ ] [ ]

[ ]

2 12 2 1 1 2

2 2 2 1 2

2 -1Q I T C ; F M L T N ;

L m C N m

= =

/(:/#MXJ$:JG')4):

)4.1([ ] a b c d e f g G M L T I N J =:C$(temprature)

N:C)(quantit de matire)J:7C4(intensit lumineuse)

01:'QMD'* #@ @@ - /'(1.

[ ] [ ] [ ] [ ] [ ] [ ]1 1 sin 1 1 log 1 8 1 1xx e x = = = = = = =

)5.1::')4:$-H):00

( )( )a

P V b RTV

+ =?$

P GGD0V :O$TC$.

7%#@a

b

R.

A:$;0

a

VG'@XJ::

[ ] [ ]

[ ]1 2

0

FaP ML T

V S

= = =

XJO:[ ] [ ][ ] [ ]1 2 3 2 20 .a P V ML T L a ML T + +

= = =

bO $RL));J:[ ] [ ] 30b V L+

= =

XJ+')RTG'@$:JXJO:[ ] [ ][ ] [ ]2 2 2 2 1RT P V ML T R ML T K+ + = = =

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Calcul des incertitudes ]TSTuv;wTx_


5

/B-I#111RH11$CALCUL DES INCERTITUDES

1/D2( grandeur physique):

+D7 "97 -EF80./*

=8:8)G:(2*9(-.!=-'.

2/5(notion de mesure):

4'88:84 /D8'!+?-. :

2:2:0=)9E:H)3)9E:I+(7

#"DGI'. +)%:")9E:H)I+

&+)9E:(7

2"DH.

8"XJDI+4'0>"4 KxB D!2 "8L*0xxIM$F$H(erreur absolue)1O:x

(5.1)0-x x =

. M +.9 8 0 2 G/>E 8 D?9

4 # F9E8:8F /89"D

I M$64(incertitude absolue)X.

(6.1)x x

88"8>""8K-. x x+x x.

0"LL9#?D%:&*98:8:

8( ), ,X f x y z=N"7!2D0%.

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6

HL9#4XO:X0&*90%dXN"X dX .

-E:28M)9E'8:!?&*91.

8:f f fdX dx dy dz y z

= + +

L9#48KXHX#:

)7.1(f f f

X x y zx y z

+ +

J:!64(incertitude relative)#48

O:J L9

x

x

O:M0&*9O:)8.1(

X dX

X X

=

3/343:!(thormes des incertitudes)

>%M$6:(incertitude absolue dune somme algbrique)

:!N!:L9#;%O' ("8"09!?.

O5 8 -'y nu pv qw k= + +?$npq#; k H('*#@uvw:D

::uvw + HR XJyD:n u p v q w = + + .

)9.1(-y nu pv qw k n u p v q w= + + = + +

%:EH)')47:

)10.1(0 ( )y y y u=

?$:$:0:yR+H:yC$:u

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7

)6.1:)M'&$J2

1 12.762=m g2 57.327=m g.1 +HR#MLD2 = m mgH$MM.

A:

2 1

1 2

44.565

4 0.004

= =

= + = =

M m m M g

m m mg g

')47H)XJM)D:

(44.565 0.004)= g

1:HRJ:

51 2

2 1

9.10m mM M

M m m M

+ = =

A(:+:(incertitude relative dun produit ou dun quotient)$6D:-:K/'"(

70:HR( ) 8 : FQ$$')#;.

'%:) ML8 n p qy ku v w=?$nq $$&k

H('*#@uvw::D:uvw.

:J-+:

log log n p qy ku v w = -,*H$:

log log log log logy k n u p v q w= + +

'%\H)4@:-:

dy dk du dv dwn p qy k u v w

= + +

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8

:HR CL'&)C4.G C4.(+ M*N^:

)11.1(y u v w

n p qy u v w

= + +

)#$QMD'@CC,;*:5')diIi5C4.GC4_L+5($R :M*N.

):/(:

)( )

u vy ku v t

=

+

#*%0:( )log log log log log logy k u v x y t = + + +

dy dk du dv du dv dt

k u v u v u v t = + +

+ +

%(%:$')0diC4.'C4.:+

dy dk dt du dv

k u u v v u v t

= + +

+ +

)12.1(t

u vy u u v v u v t

= + +

+ +

)7.1: 7) E+HR @:HRH$

2Q RI t =.A:)8:HRH$):

2 2Q R I t

Q RI tQ R I t

= = + +

+HRCMD:

2R I t

Q QR I t

= + +

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Rappel sur le calcul vectoriel yzTd{:;wTxg:TS7^|


9

VVV ==

V

1/:)grandeur scalaire(:C$:J.

)+: $4C$)$E.....

2/"3:)grandeur vectorielle(Q@N OODL$ CEO. # QMD

4/.)+::7)'$C'........

3/F3)(:)4')1.2(

:D (T4L.

:L)C4(.(

4/F36:)vecteur unitaire($(OT4D.

T4 )4T')4C$:

)1.2(

5/3L/&:)lesomme vectoriel(+$ M 4/0):0.

"3&:4/0O:

1V

2V

V

1 2V V V= +

2 1V V V= +

. .V u V V u= =

VO

')41.2

O

u

V

')42.2

/M1:11141H11$111) IIRAPPEL SUR LE CALCUL VECTORIEL

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10

AB

E

1V

2V

V

Z

C

D

E;T4C4'&$: H

)loi des cosinus( 8CJ:J$RD::.

)2.2(

$)('$(V

$:%)W')4:J$;?$4.2

O?@:JACD:

)3.2(

?@:J'@BECXJ:

)4.2(

)3.2()4.2(: E;H)loi dessinus(

)5.2(1 2sin sin sin

V V V

= =

9B:M#)2 =XJ2

22

1 VVV +=1

2

VVtg =

2

sin

sin

CD CD

AC V

CD CD

BC V

= =

= =

sinsin

2

VV= 2.sin .sinV V =

1

sin

.sin .sin

sin

2 1

2

BE

V VBCV V

BE sin sinAB

=

= =

=

2 2

1 2 1 22 cosV V V V V = + +

I[\NO4.2

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11

3+6/&:)')4$R5.2(54321 VVVVVV

++++=

')45.2

>23:T4'@DD

')4:J6.2+%42V

1V

))?$:2 1

D V V=

:')4QMD)):)( 12 VVD

+=

4/`')4O$;MD#:'D D=

F3D

:)module du vecteur(

)6.2(2 21 2 1 2

2 cosD V V VV = +

1V

2V

V

3V

4V

5V O

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12

6/F3 :)s dun vecteurcomposante(O T4 ') )4T):7R #).@)(.

*(::J( ; , )O i j

R:

C#)'C:

?$

:4$C$i

j

Q:J ')$OXOY$;:

)7.2(

*/:J:.V u V= XJ:

)8.2(.cos .sinu i j = +

T4V

:J:22 yx VVV +=

')S*:22 yxV +=

)1.2:4&$:

2

2

2

1

1

1 V;y

x

y

xV

:J( ; , )O i j

R.

'$:221221 )()( yyxxV +++=_)()x(xiV; 212121 yyjVVV +++=+=

)2.2:4+%

2

2

21

1

1 V; y

x

y

x

V

:J( ; , )O i j

R.

. . ;

. .

. cos . sin ( .cos .sin )

x x y y

x y x y

V i V , V j V

V V V ; V i V j V ;

V i V j V V V i j

= =

= + = +

= + = +

x yV V V= +

cos

sin

x

y

V V

V V

=

=

V

yV

OX

Y

i

j

xV

u

07.2:5

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13

!

cos.

sin.sin.

cos.sin

VV

VV

VV

z

y

x

=

=

=

'$:221221 )()( yyxxV +=)()x(xiV; 212121 yyjVVV +==

'2::J( ; , , )O i j k

R) CCE:(

$;:zyx VkVjViV ...

++=zyx VVVV

++=

Y

X

Z

xV

V

zV

V

r

j

i

k

8.2:

D+$):

:J:

)9.2(

:JT4:2 2 2x y zV V V V = + +

)#@$.:2 2 2V x y z = + +

!!""

!

!!""

!

""

sin.sinsin.sin

cos.sin.cos.cos

;sin.sin

cos.cos

rVVV

rVVV

rr

rVr

V

yyy

xxx

zz

===

===

==

==

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14

01: ML1 T4&LV

$0OXOY:&$'@ @@E;9.2)J:

)10.2(.cos , .cos , .cosx x zV V V V V V = = =

)UC:)11.2(2 2 2cos cos cos 1 + + =

)3.2:&%Ju),44-,10(A;u),8,610(B@ '( ; , , )O i j k

R?$u=C$.

::D $HJ:):

12VD V

=T4:DJXJ:D

.2

12

2

12

2

12 )()()( zzyyxxD ++=++= )()()( 121212 zzkyyjxxiD

uDkjiD 77.10116)4()10()0( ==++=

)4.2:*4/&$:

:

.'$ $ T4V

E; +x

y

V

Vtg = :D :

&V

$0OX:14 3,5 74,054

tg = = =

7/)produit scalaireI(:4:8:1V

2V

:$21.VV

?$:

)12.2(1 2 1 2 1 2. . .cos( . )V V V V V V =

:

)13.2(2 2 21 2 1 2 1 21

. ( )2

V V V V V V = +

:9B:)ML01

=V02

=VXJ0. 21 =VV

)ML01

$V02

$VXJ:

1 2 3 4 5(4 3 ) u; ( 3 2 ) ; (2 6 ) ; (7 8 ) ; (9 )V i j V i j u V i j u V i j u V i j u= = + = = = +

(4 3 2 7 9) ( 3 2 6 8 1) 19 14 61 196 23.60V i j V i j V u= + + + + + + = = + =

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15

1 2 1 2 1 2

1 2 1 2 1 1 2

( , ) cos 0 . 02 2

// ( , ) 0 cos0 1 .

V V V V V V

V V V V V V VV

% = = =

= = =

1):CE'F

R?$ABE;:cos..ABFW =?$);( ABF

=

):W1:8DF

AB(H):.W F AB W=F.AB.cos= &

\D)2.2():2 2 2 2 2

1 1 1 11 2 1 2 1 2 1 1 1 1

2 2 2 2 21 2 1 21 2 1 2 1 2 1 2

2 . cos( )

2 cos( ) 2 cos( )

V V V ; V V V V V ; V V V VV V V V ;

V V V VV V V V V V VV V V

= + = + + = = =

= + + = + +

(6:(expression analytique du produit scalaire)

1V

2V

?$(:J4:

2

2

2

1

1

1 V;y

x

y

xV

:J( ; , )O i j

R:

'2)dans lespace(1V

2V

4:J( ; , ; )O i j k

R:

)15.2(1 2

1 1 2 1 2 1 2 1 2 1 2

1 2

. . . .2

x x

V y ; V y V V x x y y z z

z z

= + +

1D

. . . 0

1

i j i k j k

i j k

= = =

= = =

)5.2:4 C&$ H$:1 3 2V i j k = +

2 2 3V i j k = + +

.

:

H$):8CE;:11

1 21 2

.

cos( )

V V

V V V V=

( )1 2 1 1 2 2 1 2 1 2 2 1 1 2

1 2 1 2 1 2

2 2

. . . .( . . ) . . . . . . . . . . . .

. . 0

. . . .

. . 1

V V x i y j x i y j x x i i x y i j x y j i y y j j

i j j i i j

V V x x y y

i i j j i j

= + + = + + +

% = =

= +

= = = =

(14.2)

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16

O:

11 1 2

111 2 1 2

1 2

. 3 4 3 2 9 4 1 3.74 1 4 9 3.74

. 2cos( ) 0.143 ( ) 96.2

14

V V ; V ; V

V VV V V V

V V

= + = = + + = = + + =

= = = = =

8/"3:)produit vectoriel(I(:4:48:1V

2V

T4W

(

)(.$;&H):

2121 VVVVW

=(=

F37D9BW

:(proprits du vecteur)

W

)4')4(1V

2V

OD

$)L4.W

(O4H$J:

)70.4(%:

0

1

i i j j k k

i j k ; i k j ; j k i

i j i k j k

( = ( = ( =

( = ( = ( =

( = ( = ( =

01:1 2 1 2. .sin( ; )W W V V V V = =

'@$T;/(::$

)X $T4.

1 2 1 2. .sin( . )W W V V V V = =

O

W

1V

2V

W

')49.2:48

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17

4:48)S*

2

2

2

2

1

1

1

1 V;

z

y

x

z

y

x

V

:J)#@$.'),,;( kjiO

R:

1 1 1 1 1 1

1 1 1

2 2 2 2 2 2

2 2 2

i j kz x z x y

W x y z i j k y z x z x y

x y z

+ +

= = +

( ) ( ) ( )1 2 2 1 1 2 2 1 1 2 2 1W y z y z i x z x z j x y x y k = +

CH$XJ::

)17.2(( ) ( ) ( )2 2 2

1 2 2 1 1 2 2 1 1 2 2 1 1 2 1 2. .sin( , )W y z y z x z x z x y x y V V V V = + + =

)6.2:T4H$LW

48:)2,0,1();1,1,2( 21 == VV

@Z.

:

[ ) ( ) ( )2 2 2

1

(1 2 (0 1)]. 2 2) (1 1 . 2 0) (1 1 . 2 3

2 1 1 6

W i j k W i j k

V

= + = +

= + + =

2 2

2

2 2 2

1 2

1 2

1 0 2 5

2 3 1 14 3,74

3,74

. .sin 3,74 sin sin 0,683 43,06. 30

V

W

W

W V V V V

= + + =

= + + = =

= = = = = =

9/4?( "(:(gradient, divergence, rotationnel)

I(::),,( zyxf;$#)ML),,( zyxf.:'@),,( zyxV

#)ML4;$),,( zyxV

4.

2@3(oprateur):4:%)(nabla)

1:

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18

)18.2(i j ky z

)= + +

?$

y

7 #4 :D. U 2@3MD.

?(:(gradient)#) ML),,( zyxf XJ :4 :)2:

)19.2(( )f f f

rad f f i j k y z

=) = + +

).27:UH$2 3( , , ) 3x y z f x y z= =H:kyxjzyxizxyfgrad

.3.9.6 32223 ++=

"(:(divergence)#) ML),,( zyx VVVV =

DXJ O4 :) 2::

)20.2(.yx z

VV V

divV V x y z

= ) = + +

).28:4H$:

2 3( , , ) 2 3 9V x y z xyi yz j xy k = +

A:2 22 3 0 2 3divV y z y z = + =

:(rotationnel):4)ML),,( zyx VVVV =

DOXJ:

)21.2(( ) . . .y yx xz zV VV VV V

rot V V i j k y z x z x y

= ) ( = +

CGCL'*TL$:

/:':

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19

x y z

i j k

rotV A B C x y z

V V V

+ +

= = + +

H/H$, ,C B A0:

yz

y z

xz

x z

+ i

VVA = +i

y z y z

V V

-j

VVB= = -j

x z x z

V V

C=

=

xz

x z

k

VV= +kx z x z

V V

U/'&$:J)21.2:(

y x xz z z

x y z

i j k

V V VV V V+i -j +k

x y z y z x z x z

V V V

+ +

=

).29:$T4H:kxyjyzixyzyxV

32 932),,( +=.

A:

( )

( )

2 3

2 3

( ) 27 6 . (9 0). (0 2 ).

( ) 27 6 . 9 2

rot V xy yz i y j x k

rot V xy yz i y j xk

= +

=

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20

10/:1)le laplacien(I(:().:

5;RUPD:)22.2(( )

2 2 22

2 2 2. ( )

f f ff f

x y z

) ) =) = + +

5;R"3UPD:

)23.2(( )22 2

2

2 2 2. ( )

yx zVV V

V V i j k x y z

) ) =) = + +

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Principaux systmes de coordonnes T;c_:Wx7:;WZp;]


2121

a bcdefMNO gNKLMNO hij hJ gklJ gdLJ hiiLm nopbpqKJbrstN uvwdNO xyzMNO {p{weN74:

1/G+:)inertie ou galilensrepres d().:)%L::-156451642:(

& R*8%:J6$(0E$)0O(#@$L$O64(M.

I(:%#@$L$')')4H&S(rfrentiel)(repre)H&S.

1):)0+ (3$=.I/)0+ (364D3)$=I/.

):J(C$)G)$:J.0:JR E 2M@$L 7J#@$L?;@

$:$0XJ: ( ), , ,X Y Z tMD ;@$0EKitJ|a KikK}~.

ML1),,,( tzyxOMr =0M$:Jt)$XJ:J2 P+)(trt .

2/G&%+:

^ qfd7S:(repre de Copernic))NbCopernic ( 1473-1543)(sdrtJaKwJ ljQLJgdLMNO ObbzMoMNObNKrPQJhJ

KefJbejKzolzwboeJa biM\NObpKtLj)I[\NO1.3.(nOz[dNQjKLNObikK}cNOKQMNOa nOz[NObQbO{NgdLMNOOIMLep.

KMnrsNOza{mO-btlM\NOza{ma {Oazplzt.

*:(repre gocentrique)()$?;@2$I/)

/KiO{NbiikQNObMvO IIIPRINCIPAUX SYSTEMES DE COORDONNEES

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2222

6) @ ?;@.E/ )$ MD 'I/'$&R.;* %'$I/'24.

':(repre terrestre) MD$?;@2 I/(C:D.

I/ )$/MD'.:J) I/MD.

3/().:(coordonnes cartsiennes)

O/yT9:;qfdZ:;:(repre spatial)QweMNOxyzJ{p{wmh[MpbikK}~bQwNOKO

urstNOMgdLMNOu~),,;( kjiOR

LyzJKL\jOMmKiO{ja)))Lren

Descartes :1596-1650(:D :

x:bdKcNO(abscisse)y:nimQeNO(ordonne)z:zdLNO(altitude)

xyzJKLbjKeh[MpMI[\NKj:

)1.3(. . .OM r x i y j z k = = + +

Yi

k

X

Z

O

M

z

x

yj

r

m

I[\NO2.3:bpPimK[NOKiO{O

()

M\NO

I/

6)

M.M

')41.3

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2323

A/*(:(repre plan)urstNO QweMNO xyzJ {p{wm h[Mp bpzeJbQwNO K OMgdLMNO u~

),;( jiOR

)I[\NO3.3(ieiO{jxayLyzJKL\jaOM:

)2.3(. .OM x i y j= +

?$:x:bdKcNOay:nimQeNO

/q\hxZ:;qfdZ:;:(repre rectiligne)zwMNKjKceO h[MpbMiseJbQwNOKOOXne[i:

)3.3(.OM r x i= =

r

x

y

Y

X

u

j r

Y

X

!!

4/.))coordonnes polaires(KMNOuMetphi[MpNKtzeJNhQweMdNuvwdNOxyzMNOhiiLm

iO{KjehiirsNOhie( ),r!.)I[\NO4.3.(r:ursNOQrsNO-(rayon polaire)a!:birsNObpaOPNO(angle polaire)

I[\NKjxyzMNOKLbjKeKtt[Mpi:

)4.3(.r

OM r r u= =

E;&$'@)8.2()):

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2424

.cos .sinru i j! != + .sin .cosu i j! ! != +

:J0T4))O#@$.')4:

)5.3(. .r r

OM r A u A u! !

= = +

i:( ),r A!KmKQJKMOM

{KsNOu~( ),ru u!

:D#@$. #@$.E;:

)6.3(.cos

.sin arcsin

xx r arccosr

y rr

!

!

= =

= =

5/).:(coordonnes cylindriques)

Oi KikK}~KMNOK"aozu~ OPiMJ OaqKLdmQweMNO xyzJ{p{wmhwepbiOzrO KiO{OKMLeO( ), ,z" !i:

":ursNOQrsNO-(rayon polaire):birsNObpaOPNO),oxom((angle polaire)z:zdLNO(altitude)

zZ

X

Y

"

!

O

u!zu

u"!

z

m

r

"

u

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2525

biNKeNObjKe[NObwhJswe:.OM r Om mM r u= = + =

)7.3(. .r u z k "

"= +

')45.3+$:

)8.3(.cos .sinu i j" ! != +

*M$u"

r

u

.I[\NKjxyzMNOKLKbjKeOKtt[Mp:

)9.3(

. .cos . sin .

. . .

OM r i j k z

OM r i x j y k z

" ! " != = + +

= = + +

KL\NOKIpzwmh[MpKMOM

I[\NOdz[pibiOzrOKiO{ON:

)10.3(. . .z z

OM r A u A u A u" " ! !

= = + +

i:( ), ,

z

A A z" !

=

KQJuOM

{KsNOu~( ), ,

z

u u u k " !

=

.'&$C$T4Cu

!

CO:%)( ), , zu u u k " ! = XJO Cu

!

:48Dz

uu

"

.ML:

)11.3(.sin .coszu u u i j! " ! != ( = +

u~ himKLNO bsjKrMj)7.3(KiO{O a bpPimK[NO KiO{O hij blLNO ete

biOzrO:

)12.3(

2 2cos

sin /

arccos / arcsin /

x yx

y arctgy x

z z x y

"" !

" ! !

! " "

= +=

= =

= = =

bvlJ:KO0=zbirsNOKiO{OdQLe.

6/).:(coordonnes sphriques)

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2626

zsmO{LNOahObpaQ[NOKiO{OKMLeO~PiMJa{j)Zr(I}cMNO-p:i:

r:O QrsNO-ursN)(rayon polaire(Z:M(azimut):QLNOKMm.(coaltitude)

NOKiO{Oa bpPimK[NOKiO{OhijKlLNOhJKi{tswebpaQ[:

)13.3(cos sin cos

sin sin sin

sin cos

x x r

y y r

r z r

" ! !

" ! !

"

= =

= & =

= =

)14.3(

2 2 2

arccos

r x y z

zr

yarctg

x

!

= + +

=

=

O hijblLNOKJO a bpaQ[NOKiO{u~biOzrOKiO{:

)15.3(

2

sin

cos

2

r r z

z r arctgz

" "! ! ! !

"

= = +

= =

= =

u~xyzMNOKLne[ObimK[p{NOKiO{:. . .OM r x i y j z k = = + +

bpaQ[NOKiO{Ou~KJI[\NOdne[i~:

)16.3(. . .r r

OM r A u A u A u! !

= = + +

i:( ), ,r A A! KQJuOM

{KsNOu~( ), ,ru u u!

W_:KiO{KjOQcNOIbireNbpaQ[NOQimIs:

rhJ0NZhJ0N2hJ0N.

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2727

X

Z

Y

mx

y

z

O

r

4)'7.3:.$@#)4)'8.3EC.$@#)

O

Z

X

Y

r

M

m

u

u!

r

u

!

uTz];WdV:;;c_:

)17.3(

.

. .cos sin.cos .sin .sin .sin .cos

. . cos .

.sin

r

r

r r u Om mM

Om u i ju i j k

mM z k k r

r

"" " ! !! !

"

= = +

= = + = + +

= =

=

.sin .cosu i j! ! != +

C( ), ,ru u u! {OzNOKL~{JKLeJu

hijuKL\NOO{oNOIKz

u!

au:

)18.4(.cos cos .cos sin .sinru u u i j k ! ! ! = ( = +

7/).:(coordonnes curvilignes)O O%6$0 $ )

&%$(abscisse curviligne):5O5@*O.

(N$@$.2Ps:OL:

)19.4(OM s=

MO

')49.3:$#@$.

+

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Caractristiques du mouvement W^7g:;];nZ


2929

Wn:;]vYTdZ:;:(quation horaire))M:J)(repos)#@$.#) ML, ,x y z

) :J)$(mouvement)@$. QMD #$& ML0 #. 1:

(2.4)( ), ( ), ( )x t y t z t

Oa{NOuMbitJPNOqKLMNOI[\NKjKtQiLeNOh[Mpa bQwdN:

(3.4)( ), ( ), ( )x f t y g t z h t= = =

uTxZ:;:(trajectoire)

KJbKLeJbtJ|lKedeOueNObiNKeeMNOxyOzMNOzMoJzbpqKJbrs.KpqKJz[pKMdNh[Mp)pQrNO(KiMaa)QMsNOKJ.(

gdLMNOu~bdireMNOKiO{KjgembpzeMNObQwNObOq-piKMhieiO{jQLJxyzMNO:)(),( tytx.

bNO{NO)(xyx Mm:;W:YTdZ:;TuTxZf:WXnu.(quation cartsienne de la trajectoire)egh:YTdZ:;STn:;|gSuTxZ:;W:YTdfzhn:;.

T1.4:u K}cNOu~~bpqKJbrsbQwNbitJPNOqKLMNO:)MoNOu~O{zNOIdbiNa{NOb(.1/d[KJKMdNbpPimK[NObNqKLMNO{a2/bvwdNOu~xyzMNOKLKne2t s=.

w;:;:1/QfebN{jhJPNOxKu~zLgzKMNObNqKLJdI-wt~~K[JxrhKza.

2

22

1.25. 2.

xx t t

z x x

= =

= +2/xyzMNOKLK:

2( 2)(2 ). ( 5 4 ). 4 12tOM t i t t k OM i k == + + =

T2.4:KOgdLMNOu~b~QLJbpqKJbrsbQmK[p{NOuMNKjhieitJPNOhieNqKL:

)cos(

)sin(

!* +=

+=

tay

tax

xeMNOKMNOI[zKM~

),;( jiOR

2

42

x t

y 0

-5t t

=

=

= +

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3030

w;:;:-QkOqbNqKLJdI-wt~QrNQKMLMoghieNqKLMNOxjQKQra:

2 2 2

2 2 2

2 2 2

sin ( . )

cos ( . )

x a tx y a

y a t

* !

* !

= + + =

= +

4/"F3:(vecteur vitesse)hJPNO{albzrsMNOb~KMNOubQNOQeL.

W[UhZ:;Wz7x:;Td:(vecteur vitesse moyenne)

I[\NOl2.4:bvwdNOhijtxyzMNOQweMNOKi~I\pueNObvwdNOa'tueNO

xyzMNOQweMNOKi~I\p'biNKeNOKLNKjQLJbrzeMNObQNOKL~:

(4.4)

'M

KseOKLMp.

F30":(vecteur vitesse instantane)T42 :JT4($$t

O(3(drive)0T4:

(5.4)t

dOMv

dt=

%:T4)(tv:JO$MQ$ 7O

)$)')43.4.(6M04C4C():J

+4:

(6.4). . . . . .OM r x i y j z k v x i y j z k= = + + = + +

''

'moy moy

MMMMv ; v

t t t= =

''

'lim lim

'

t t tt t

OM OM OM dOM v

t t t dt

= = =

KMNOM

M

(t)v

T

O

')42.4

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3131

9)conventions((()Newton(:4L2$0

GL(M.*=G(/4#)ML;0DXJ'GL(M2$.

(()zLeibnit(:G4LyhJPdNbtNKj1:dt

dy.M)D

H)):

0"F363:(module du vecteur vitesse instantane)(7.4)

:JC$KSu:1./ = smsm

4#@$LOMv:

5/F(F3:(vecteur acclration) ';*GTC$.

UhZ:;uTxh:;Td:(vecteur acclration moyenne)ML%*$ta't:40OM'OM

$:4v'v)')44.4(2TT4XJC:

)8.4('

;'

moy moy

vv v v a a

t t t t

= = =

dt

dzz

dt

dxx === ;

dt

dyy;

2 22v x y z = + +

Rz

y

x

R vz

vy

vx

v

z

y

x

OM

=

=

=

v

')43.4:$T4

M

O

M v

'v

moya

')44.4

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3232

ygf:;uTxh:;Td:(vecteur acclration instantane)4 O2P )$:$TT4(drive)T4

$:

(9.4)

\)()$ C4/2*#E; C) '):

2 2

2 2 2

. . . . . . . . .

. . . . . .2

OM r x i y j z k v x i y j z k a x i y j z kdx dy dz d x d y d z

v i j k a i j k (10.4)dt dt dt dt dt dt

= = + +

= + +

= + +

= + + = + +

%:TT4) $7)')45.4.(

0

F(

F3

:

(module du vecteur acclration instantane)TT4C4H$C:

(11.4)

91B::J):)):

(12.4)

2 22a x y z = + +

. . . . . . . . .

x x x

y y y

R z z zR R

x y z x y z

x v x v ax

r y v y v a y v a

z z v z v a

OM r x i y j z k v v i v j v k a a i a j a k

= = =

= = = =

= = =

= = + + = + + = + +

dt

OMd

dt

vd

t

v

tt

vva

2

tttt 2''lim

'

'lim ==

=

=

2

2dv d OM a

dt dt= =

)(ta

')45.4:TT4

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3333

(:)$)"() ML. 0a v

)D() ML. 0a v

.P)$QJOg'T4Qv.

)3.4:D 0T4) ML

=

=

=

3

2

54

2

tz

ty

tx

OM.T4 T4 L

@$T')C4H$.

A:L'&J+4::

2

4

4 . 4 3 6

16 92 2

v t i j t k a 4i 0j tk

v 16t t ; a 16 36t

= + + = + +

= + + = +

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Mouvements rectilignes WZ\hxZ:;]T^7g:;


34

1/0((:(mouvement rectiligne uniforme)I(:T4 D) ML)$:J)

T4XJ: @. :$ *OXukO{ejO Q\NO q{w a gdLM:

0;0 xxt ==

E;&%CL'&) 2R:

/:D '&$C**=:J:

(13.4)

:x$&%0xbikO{ejObdKcNO.

B:(diagrammes du mouvement) T'):'@:D)$#*

R'.)')47.4.(

t0a=

t

tev C=

t

0 0x v t x= +

0( 0)x =

t

0 0v t x= +

O OO O

0 0

0 0 0

0 0 0

. .

0

x t

x t

x t

x 0

dxv x v dx v dt dx v dt

dt

x v t x-x v t

= = = = =

= =

+ +

0 0.x v t x= +

. ..XO

0

0

=t

x

t

x

')46.4:)$

/B-IV#)$MOUVEMENTS RECTILIGNES

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35

)4.4:$ #R:D ):0;42;2 =+== ztytx)#$'):J.()$D.

(:0=,z)$.)ML0;0 ==, zy*)$.)ML0;0;0 $$$, xyz)$7J.

:J?$6M'h)$RD:4y x, = +)$ML.

@ ) R )$ ) $: T4 H)C4H$@0T4+4:

ML)$.

2/6G((0()mouvement rectiligne uniformment vari(I(:) ML CG )$ )

T @.":7.4 :0vvt == ;0 E;

H)2:tvvv

v

t

atvadtdvadtdvdt

dva

0 0

00

==== + +

'&$*/:J/:D $:

(14.4)0

.0

v v t v= +

::J * ML0xx;t == 0+ E;

H):+ + +=+=+==x

x

t

0

0)0)

0

dtv(atdxdtv(atdxvatdt

dxv ))0

:DXJO:

(15.4)20 0

1

2x at v t x= + +

B:

:J$;')48.4'. T'))$#*.

2 2 12 2 2 2 8 2.83v i j v v ms= + = + = =

t

')48.4:)$#*

O

a

a= Cte

v

tO

0vatv +=

00 =v

tO

x

0x

00

2

2

1xtvatx ++=

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36

+%+A:)(2 02

0

2 xxavv =

$(:)$)(acclr))ML0.

>va

)$)7(retard)) ML0.

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37

3

0

14

3v t t v= +

J$C'&$')';:4 2

0 0

12

12

x t t v t x= + +0

0

t

x vdt= + +

&%')$\:0x0v7R.#H$L'&:JI3t s=:

1

0

33 1

40

t s x m ; v ms= = =

$'. :H)*/:J:

314 1

3

v t t= 41 3

12 4

2x 2t t t= +

4/W:;WZ\hxZ:;W^7g:;:(mouvement rectiligne sinusodal)X7d:bitJPNObNqKLMNObjKeh[JObpqKJbrstNbiibMiseJbQwNOz[m

I[\NKjKeQwN:(16.4).cos( . )mx X t* != +

ea:).sin(. * += tXx mx: uvwdNOKrMNOabdKcNO(longation ou abscisse instantane)

mX:uMvOKrMNOabLNO(amplitude ou longation maximale):Qiep'

$E:

*:bQwNO(pulsation du mouvement)!:bikO{ejObwc-NOaukO{ejOzrNO(phase initiale))+t: (bivwdNObwc-NOauvwdNOzrNO(phase instantane).

1)cos(1 ++,+ !*tXxX mm

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38

":+4:dxv xdt

= =

(17.4). sin( )mv X t* * != +

$EQMDG:

. . 1 sin( ) 1m mX v X t* * * ! + , + +

F(:+4:dva x vdt

= = =

(18.4)2 cos( . )m

a X t* * != +

QMDGT$E:

2 2

m mX a X* *+ / /

:$TC)):

(19.4)2

.a x*=

Q.:JO) '0HT.06$T))'&%(

:/O6$W).

('2(quation diffrentielle du mouvement)E;))T:

(20.4)2

2

2. 0d x x

dt*+ =

2 2. 0a x x x x* *= = + =

% QMD'$ ')4Dcos sinx A t B t* *= +(M $'%@@#;')4O)):)cos( !* += tXx m

% @ $'mX' ') 7. 4 J0x0vY#M '&$?$h7R

mX.

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39

!

!

sin

cos0

0 m

m0

Xv

Xxt

=

==

B:')4'@10.4')#* '. T)$

)*0=!.(

x=f(t)

a=h(t)

V=g(t)

2T3T/4TT/2

-Vmax

-Xm

+Xm

+Vmax

+amax

-amax

t(s)0

X,V,a

).47:'@:D:)5.01.0sin(4 += tx):J ')#$MKS.(

:/. $%& )$7. H/T .U/7.4./:J T0st 5=.

1D/)$#*.

:/)$7.$%& :

H/H$T :

U/$7.4:

0.4cos(0.1 0.5)v t ; a -0.04sin(0.1t 0.5) -0.04x a=-0.04x= + = + =

')410.4:)$#*

2

24

11.59.10 ;

mX m ; T T 20 62.8s ;

N N Hz 0.5rad .T

*

!

= = = =

= = =

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40

0

1

0 4sin 0.5 1.92 1.92

0.4cos0.5 0.35 0.35

0

0 0

t x m x m ;

v ms v m

= = = =

= 0 =

/:J T05t s=

:

D1/)$#*: HY&8%).L.

t+3T/4t+T

t+T/2t

a=-0.04sin(0.5t+0.5)

v=2cos(0.5t+0.5)

x=4sin(0.5t+0.5)

x,v,a

t(s)

-4

-2

2

4

0

011.4:"!99E

5 :

.

-1

-2

t s x 4sin(0.5 0.5) x 3.36m ;

v 0.4cos1 v 0.22ms ;

a -0.04sin1 a 0.034ms

= = + =

= =

= =

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Mouvement dans le plan WXhxZ:;]T^7g:;


41

#@$. 6$ 0 $ ) L ) ML#@$..

1/).:5(&':)D:$)C.(

)#@$.6$0

D +): (21.4)

J#@$.:

22.4)(

?$:.cos .sinru i j = +

::

( .cos .sin )OM r r i j = = +

(:r:( )r f t=( )t =.

6"":)#@$.:

(23.4)

.#@$:')4LL12.4C$:4:))ru

uC$:4Ri

j

:

C

O

Y

X

v

v

v

u

ru

i

j

r

')412.4:@$.

OM r xi yj= = +

.r

OM r r u= =

v r xi yj= = +

C-IV/#)$MOUVEMENT DA S LE PLAN

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42

(24.4)

4H$+O:

)5.42(

.sin . cos .

.cos . .sin .

r r

r r

du d d d du d i j =u . =u .dt dt dt dt dt dt

du d d d du d i j u u

dt dt dt dt dt dt

= +

= = =

#@$.C\H$:

)6.42(

)(v

Ev:

F(6":#@$.:a v r xi yj= = = +

#@$.:C+4:

. . . . . . . .rr

dudua v r r u r r u r u

dt dt

= = + + + +

.( . ) . . .( ) . . . .r r

d da r u r u r u r u r u

dt dt

= + + + +

QMDT47CL'&' C#@$.T:

. . . . .( ) . . . .r r

a r u r u r u r u r u

= + + + +

)7.42(2( . ). (2 . . ).r

a r r u r r u

= + +

)TM$;raa

:

)8.42(r

a a a

= +

:JO4:

)9.42(2 2 2( . ) (2 . . )a r r r r = + +

.cos .sin .sin .cosr

u i j u i j

= + = +

r

r

du drv r r u

dt dt

= = +

r

dr dv u r u

dt dt

= + &

. . .

rv r u r u

= +

2 2.. . .

( . ). .

r rr

r

v = r uv r u r uv r r

v r uv v +v

= + = +

==

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43

9B:D:(mouvement circulaire)r R Cte= =)Cte=#@(T4XJ:

)30.4(v R u

=

TT4C:

)31.4(2. .. . .ra R u R u = +

$;)T:0F((acclration normale)O$(M$OD

C7))(aG43D":)32.4(

F((acclration tangentielle)(M:JO$

G43D"63.

)33.4(

9B:0(D:(mouvement circulaire uniforme)@C47)$:J. r R= =XJ#@

:

)34.4(v R R *= =

2 ?$*C$ ';* $ '@ :D @$ )1.rad s.(

JT:

)35.4(2

2 2 2 .r N N r v

a a a R R a R uR

* *= = = = = & =

2/(404F( ")Frenet(:

M*D)$\)C(

) 2)$')4 T

D :J T4'$v$ N($T.

)TuNu

+J C$ :4MTN: . ')4 $;:

)36.4(. Tv v u=

TT Na a a= + XJ::

)37.4(. .T T N N a a u a u= +

2 2

N r r r Na a R u a a R = = = =

T Ta a R u a a R

= = = =

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44

N

T

( )C

v

Nu

Tu

Ta

a

a

4)'13.4:TJ:J#

NJ+:

22 22

2. .

T

T N

N

dva v vdt

a v u u a vR Rv

aR

= = + = +

=

:)36.4()37.4( #J :J T :)$M).

KOdsOzz xyzMNOKLup{NOhJz[i~ Q-tLNOKse:

)38.4(.Tr u ds= +

).48:

#@$l(:2.cos2

a!

" =

?$a#@.C4I%v0HQMD":v k"=?$0k >.

)H$v"v!T4.

:2:. .v u u v v v" ! " !" "!= + = +

J$D$Rr1"1!)2$%$R ML!!.(

E;#$#:

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45

2 4

.cos( /2).sin( /2). .

cos ( / 2) cos ( / 2)

a d d d av v

dt d dt " "

" " ! ! !" !

!! != = = =

.v!

" !=

-! $HM E;2 2 2v v v" != +

):2

2 2 2 2

4. .

cos ( / 2)

av k k"

!= =

O:

2 2 2 2 22 2 2 2 2

4 6 4 2

.sin ( / 2) sin ( / 2). . . 1 .

cos ( / 2) cos ( / 2) cos ( / 2) cos ( / 2)

a a ak k

! !! ! !

! ! ! !

= + = +

O:2 2 2.cos ( / 2) .cos( / 2)k k! ! ! ! = =

I!:JJ$:v"v!HDL'&:

2

. .sin( / 2).sin( / 2)

cos ( / 2)

a kv v v" "

!!

!= = a

.

cos( / 2)

a kv

!=

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Mouvement dans lespace mT9:;yW^7g:;


46

HG :J ' @;@ (M 8% :J )$ )#@$. /#@$..

1/).:(tude du mouvement en coordonnes cylindriques)5(&':)')414.4(

X

Z

Y

"u!

u"

zu

mO

!

"

z ( )

( )

( )

t

OM t

z t

"

!

cos . sin .

sin . cos .

z

u i j

u i j

u k

"

!

! !

! !

= +

= +

=

!

mXOY

, , )z(u u u" !

( , , )i j k

"

"

!

)40.4(. . zOM Om mM u z u""= + = +

)41.4(. .cos . .sin .OM i j k z " ! " != + +

7J)CQMDH$;C)6.3(.'RC(&:

)42.4(2 2 2 2 2ds d d dz " " != + +

5(":#@$. O 6$ 0 T4 +4 )

/.2&XJ7)$$)LDO

#)$8%:J /D-IVMOUVEMENT DANS LESPACE

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47

:":0\D. 6M)'zu k=)#@

u!0G.

C)M)5.42(4&* ')du

dt

"du

dt

!)):

)43.4(. . .v u u z u" !" "!= + +

#)@;@$R:;( )rv'"( )v!

"( )v

.

H$/#@$.C4C:

)44.4(

5(F(:&T4C'&$+4R:

C ')5.42(7C'&$:

)45.4(2( . ). ( . 2. . ) . za u u z u" !" " ! " ! " != + + +

C%)):')4:

)46.4(2 21

( . ). ( . ) .d

a u u z udt

" !" " ! " !"

= + +

)ML0z=R Cte" = =C)31.4()$T7.

#)?;@6M)T:;( )ra

'"( )a!"( )a

.

2 2 2( . )v z" " != + +

. . . . . . . . . zdu dudv

a u u u z udt dt dt

" !

" ! !" " " ! " ! " != = + + + + +

. . z zdudr d dz

r u z u v=r= u u

dt dt dt dt

"

" "

"" "= + = + +

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48

2/).:(tude du mouvement en coordonnes sphriques)5(&')')415.4(

MD:JE;26$0XJ :

)47.4(. rOM r r u= =

( )

( )

( )

r t

OM t

t

!

m)ME;#13.414.4C4( , , )r

u u u !

( , , )i j k

:

sin .cos . sin .sin . cos .

sin . cos .

cos .cos . cos .sin . sin .

ru i j k

u i j

u i j k

!

! !

! !

! !

= + +

= +

= +

u

u!

r

u

!

C(&'.:

)48.4(

RL 2$%$R H: $R( , )OX Om! =

( , )OZ OM =MD)S*0:JE.

5(":C+4T40:. .rv r r u r u= = +

T4+4u'&$JCC@:

.cos cos cos sin sin sin .sin cosru u

u i j k i j

!

! ! ! ! !

= + + +

2 2 2 2( sin . ) ( )ds dr r d rd ! = + +

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49

(:. sin .ru u u ! ! = +

IT4CL'&:

. . ( sin ) .rv r u r u r u ! != + +

T4@;@)#):

)49.4(sinr r

dr d d v v v v v u r u r u

dt dt dt ! !

!= + + = + +

( , , )ru u u ! 04:D C4 CC:

#R J( )r t( )t( )t! L '& Y

rvvv!4 $ @ T4 ) #)T

.

5(F(:TCL'&+4R:

T4+4::J:A-"QMD)N:

)50.4(

2 2 2

2

( . . .sin ).

( . 2 . . .sin .cos ).

( . .sin 2 . .sin 2 . . .cos )

ra r r r u

r r r u

r r r u

!

!

!

! ! !

= +

+ +

+ +

6M) D #R J( )r t( )t( )t! L '&

raaa! TT4)#):T4$a

.

. ( sin ) . .rdv d

a r u r u r udt dt

!! ! = = + +

2 2 2r ra a a a a a a a ! ! = + + & = + +

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50

).49:)$M0T4:)/#@$.:JJ

OM

?$:. . 2OM a u bt k ; =ct " = + , ,a b c#@

.1/RT H$.2/E2&H$$.8'$)COZ.

:1/H$v0T4+4:

( )2

.2

2

du v a u bk dOM d v au btk v a bk v actu bk

dt dt dt ct ct

"

"

= += = + = + = +

= =

T4C4:2 2 2 24v a c t b= +

T4+4TH$:

2

(2 . ) 2 ( . ) 2 . .1

2 . . 2 .2 . 2 2 .

dudv d d a act u bk a ac t u a ac t u

dt dt dt dt

a ac t u u a ac t ct u u a ac ct u u

" " "

= = + = = +

= + = + = +

TT4C4:2 22 4 1a ac a t = +

2/8$.E2&H$)((rayon de courbure):

;CH$$:22

2 ct tc c

= = = =

@8$.E2&H$Q :T:JI:): A$%=A-"!!!(.

22 2

2 2 2 2

2 2 2 2 2 2 2

4 16!!!!! 2

4

N T

N

T N

vR ; a a a ;

a

dv a c t dv a c ba a a acdt dt a c t b a c t b

= =

+

= = $ = =+ +

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51

( )

( )( )

32 2 2 22

( ) 12 4 6 2 2 4 2 2

32 2 2 2

12 3 2 2 2

4

2 (16 4 )

82

2 (128 1 16 )

t

N

a c t bvR

a ac a c t c b t b

a c t bRc ac a c b

+

= =

+ +

+

= + +

).401:6$DE2&C)YR:

( , ) 30 ; ( , ) 2OZ OM OX Om =at != = =.

1/ TH$R.2/K)$ D

:1/H$vTaR:

ML1, , r! )#@$.XJ)#@$.L:D:

C4@)#@$.#)H$+4R:2

2

2 2

sin cos cos cos sin sin sin( )

sin sin cos sin sin cos cos( )

cos sin 0 0

x x

y y

z z

dxv x r r r v Rat at

dt

dyv v y r r r v Rat at

dt

dzv z r r v

dt

v x y

! ! ! !

! ! ! !

= = = + & =

= = = + + & =

= = = = & =

= +

2

z v Rat+

=

@ ) H@$. T #) H$ @ C +4R

TC4L'&:

2 2 2

2 2 2

sin( ) 2 cos( )

cos( ) 2 sin( )

0

x

2 2y

z

a x Ra at at at

a a y Ra at at at a=Ra 1+4a t

a

= = +

= = +

=

sin cos

sin sin

cos

x r

y r

z r

!

!

=

=

=

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52

2/)$ ::D :2 22

x y+ =32

z R=

DE2&C7XJ2

D)3(0,0, )2

T:#@2a!, = =7)$.

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Mouvement relatif bitNObQwNO


53

1/&G(::

E+ ') ( % ) )$ 0)M*6$0.

:#)$')D\$) L D. :D J6$*=6$$;)'$)2) K%D

K*n)$:J$%*

'E

*

H$

6$

%

T

0

2*HE.

1):$&R:(EO)4 -)$:

()cyclode.($:(7 )$.

E 'E #$; :D 2) J ) D/ **n$)$:J%.

2/"(:)B)$:J OXYZ.$;I%:JO.')416.4

O

X

Y

Z

B

BAr

Ar

Br

BV

AV

BV

ABr

AV

BV

$;O:D:AAdr

Vdt

=

P21 1:

ABAB

drV

dt

=

?$AB A Br BA r r = =

.O:

111111)1$1/E-IVMOUVEMENT RELATIF

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54

)52.4(AB A BAB AB A Bdr dr dr

V V V V dt dt dt

= = =

B$;O:D:BBdr

Vdt

=

P21 1:

BABA

drV

dt=

?$BA B Ar AB r r = =

.O:

)53.4(BA B ABA BA B Adr dr dr

V V V V dt dt dt

= = =

'AB BAV V=

o(1B)B1.

'&$)$:')+4:

)54.4(AB A BB AB A BdV dV dV

a a a adt dt dt

= = =

)55.4(BA B ABA BA B AdV dV dV

a a a a

dt dt dt

= = =

D 'AB BAa a= (oT1 (

)TB1.

).411:1/6$BEP+:1110 .km h

190 .km h:.$4T11B$:J:/QR%:J

H/)D:J.2/#)%

30:DJCBCA.

:1//C C B:D:AB A Bv v v=

.T4

C$e% XJhQR)')417.455(Q(e

::1110 90 20 20 .AB A B AB ABv v v e e v e v km h

= = = =

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55

Av

Av

Av

Bv

Bv

Bv

BAv

30

e

e

H/ \)) QR)')417.45H5(XJ:

( ) 1110 90 200 200 .AB A B AB ABv v v e e v e v km h

= = = =

2/D 30)')417.45U5(

( )

( )

1/ 22 2

1/ 22 2 1

2 cos30

110 90 2.110.90.0,87 54,5 .

BA B A BA B A A B

BA AB

v v v v v v v v

v v km h

= = +

= + =

$ $ABv:%)6M E+

H:90

sin sin30 sin .0,5 0,82 55,1sin30 sin 54,5

BA B B

BA

v v v

v

= = = = =

C:JH):MDCS(154,5 .km h

)')4H$17.45U5(L55,1.C:JH)BSCA (154,5 .km h O L )

( )180 30 55,1 94,9 + = .

%D *=6$6$QMD#).)2)J$;)'$%*MDK*n)$:J$

H::JO.

3/ 9:( )a( )rE$N$').

L')418.4.Ra+(repre absolu))Q.R::

(repre relatif)

Q)$+. :(point matriel) )$:J.

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56

4)'18.4:+:

yi

k

x

z

O

j

Z

Y

X

M

A

(Ra)

(Rr)

.'j

'i

'k

OE'$;HE').:':J7QMD0:

$;:J( )a

, ,i j k

:J@Ra

:J( )r

', ', 'i j k

CG'( )Ra

0r OM=

'r AM=

a

drv

dt=

'

e

drv

dt=

Taa dvadt

=

ee dvadt

=

% 01:+ :J J't t=$; ( $;)$+R :MD %;.

MD 'Q% ) -.M ) RRCG$:JRLRIJ

:$$:JQM*2DMD 8.

';1:

')4$;18.4:

)56.4(OM OA AM = +

( ). . . ( . . . ) '. ' '. ' '. 'A A AOM OA AMi y j z k x i y j z k x i y j z k

+ + = + + + + +

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57

(";1:+4C)56.4(E;'&$#2*:

' ' ' ' ' '' ' ' ' ' '

dOM dOA di dj dk dx dy dz x y z i j kdt dt dt dt dt dt dt dt = + + + + + +

)57.4(

' ' ' ' ' '' ' ' ' ' '

a e r

dOM dOA di dj dk dx dy dz x y z i j k

dt dt dt dt dt dt dt dt

v v v

= + + + + + +

av

:(vitesse absolue)(( )Ra.ev

:(vitesse dentranement)6$(( )Rr

+( )a ) ) Dav6$u~( )Ra#) ML

#@$LMu~( )Rr) ML ( @M )( )r:

0r e av v v= =

rv

: (vitesse relative) (:

( )r. ) D )av6$u~( )a ) ML( )r

)( )a:0e r av v v= =

: @;@#E;"A(;:D:

)58.4(

01:

) ML( )a( )r*n $ ))0ev =

(E XJ % :4 - % :0

%*)OM OA$

.() ML( )r$ )$ :J) R( ( )Ra?$

', ', 'i j k

XJ@evM.

' ' '0 e

di dj dk dOAv

dt dt dt dt

= = = =

a e rv v v= +

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58

"(;1:+4C)57.4(E;L'&2*#

:

)59.4(

aa:+T(acclration absolue)TD( )a.

ra

:: T(acclration relative) T D

( )r.

ea

:T(acclration dentranement)TD( )r

( )Ra.

Ca

:T:)!F((acclration de Coriolis)'LO1832)Gaspard Coriolis 1792-1843.(

!F(:

)MLM)( )r:

)ML( )r$)$:J)CG $(( )Ra:

2 2 2 2

2 2 2 2

' ' '0

' ' '0 0

e

a r e

c

d i d j d k d OAa

dt dt dt dt a a a

di dj dk a

dt dt dt

= = = = = +

= = = =

).421: E4 @ #E 18ms.#E QMD H (N

/UC:1

50 .km h

K

2 2 2 2 2

2 2 2 2 2

2 2 2

2 2 2

2

' ' '' ' '

' ' '' ' '

'. ' '.2

aa e

r

dvd OM d OA d i d j d k a x y z adtdt dt dt dt dt

d x d y d z + i j k a

dt dt dt

dx di dy d +

dt

= = = + + +

+ +

+

2 2

' '. 'C

j dz dka

dt dt

+

' ' '0 0C

dx dy dz a

dt dt dt = = = =

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:

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av

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v

1 150 . 13,9km h ms =

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)62.4(' ' '

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r r x i y j z k v i j k dt dt dt

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.k* *=

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dvdv dvdv di d j dk i j k v v v

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)69.4(( )rdr

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64

I'5)59.4(.R58K:

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d OA d di dj dk d OA d a x y z r

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65

)73.4(( )2 2 2

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66

( )

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e e

z

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dtu u u

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' . .

' 0 0

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Lexique Arabe-Franais 9yS7z-yx7


67

9LEXIQUEArabe-Franais * yS7z-yx7 /I

Franais "

instantan QInitial D(Frottement 5(Coordonnes )=Rduction (BTerrestre '+Incertitude A(=Fondamental +Puissance "(Stabilit (Continuit (Cylindre Collision 9Convention R19Artificiel "9Maximal 0"+Horizontal +Cintique Universel Ampre +Tube A+Conservation 02Courbure Glissement >:Translation A

Oscillatoire *(%ABar Tonneau Simple Dimension

Divergence "(Roulement 4 ?()(Gradient ?(

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68

Giration (Rotationnel (Gravitationnel *(Quadratique (Ordonne

A((Nutation (+ SAcclration F(Equiprojectivit :*(Classification I9(Application >(Diffrentiel '2(Classique *(Contact !1(

Coaltitude K(Symtrie 0(Remarque (Equilibre (Harmonique (

TConstant(e) )Secondaire *)Seconde )

?Pesanteur $Algbrique Produit Paroi Entranement Sous multiple Particule

Atmosphre Sinus ACosinus (ASinusodal

RVolume Volumique Mouvement

Champ Hlicodal

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69

Anneau U

Extrieur Bparticulier 7B

Proprit 9BErreur PB

Circulaire *DIntrieur BPousse Exponentiel +Degr Temprature 6Degr kelvin 2Degr centsimale ou Celsius !+DRotation Gravitation

$Propre ($

Principal D

Liaison 4 Raction 2

Angulaire *)6(Angle Temps Horaire

!

liquide Drepos Vitesse arolaire V"Surface VAmplitude )0"+(Capacit )2)(Scalaire permittivit Azimut

Vertical ;3

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70

WPseudo force 6;3Intensit 63Vecteur (4 F3

Figure 37Choc 9phase +29Plaque 29Solide A9

KPression G'Lumineuse

D'Energie ;Prcession 6+IPhase 294 Longueur Module !4 634

0Tangente 0Cotangente (0

FExpression 6"Incompressible ("Moment "Inertie "Arc sinus !"AArc tangente 0!"

Arc cosinus !"(AArc cotangente (0!"Relation ;1"Dynamique 5("Statique ("Cinmatique "Altitude "Travail "Colonne "

Elment 9"

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71

XGaz 8Galilen 8

IAbscisse 9Torseur /(0+(Espace 'Action Effectif

>Loi ;Projectile 2$;Disque

7;Barre >4 A';Polaire ;Diagonal *;Fond ;Satellite "9;Candela ;Force 6;Mesure !;

5Cartsien (4 *(Masse (Densit )Sphre 6Potentiel Quantit Electrostatique /4 /Cosmique Kilogramme 8

Instantan Q4 0Cyclode Mou

Fluide &DInclin D

Hydraulique DMatire 6

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72

Matriel *Manomtre (Principe +Retard Y(Mobile 5(vecteur (Mtre (Superpos A(Equiprojectif :*(Perpendiculaire *"4 (Variable G(vari G(Compressible (

Moyen (Somme FCreux IRsultante 9axe Diagramme BRfrentiel &Composantes Centre

Barycentre (Elastique Aire Trajectoire Rectangle (Rectiligne (Plan (Drivant >(3Drive (3Plane V29matrice 29Plein 9Multiple I"'Elongation Exerc >Absolu >Equation Oprateur Isol

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73

Repre Notion /2Grandeur Baromtre G'!Presse !Condensateur 2)Caractristique Discussion 3;Uniforme 0(Courbe -Curviligne Parallle *Communicant 9

Position &;4&'Mole Mcanique 5

Normale 0Pulsation KRelatif Rayon ;I9Systme 0Thorme 0Permittivit $2Point

%Gomtrique %

Unitaire 6Unit 6

Vase "

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Lexique Franais-Arabe 9yx7-yS7z


74

Franais-Arabe * yx7-yS7z /II

Franais "A

Abscisse 9Absolu >Acclration F(Action Aire Algbrique Altitude "Ampre +

Amplitude Angle Angulaire *)6(Anneau Application >(Arc cosinus (A!"Arc cotangente (0!"Arc sinus A!"Arc tangente 0!"

Artificiel "9Atmosphre axe Azimut

BBar Baromtre G'!Barre A';>4Barycentre (

CCandela ;Capacit Caractristique Cartsien (4 *(Centre Champ Choc 9

Cinmatique "Cintique

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75

Circulaire *DClassification I9(Classique *(Coaltitude K(Collision 9Colonne "Communicant 9Composantes Compressible (Condensateur 2)Conservation 02constante ))6(Contact

!1(Continuit (Convention R19Coordonnes )=Cosinus (ACosmique Cotangente (0Courbe -Courbure

Creux ICurviligne Cyclode Cylindre

DDegr Degr centsimale ouCelsius

!+D

Degr kelvin 2

Densit )Drivant >(3Drive (3Diagonal *;Diagramme BDiffrentiel '2(Dimension +Discussion 3;Disque 7;

Divergence "(Dynamique 5("

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76

EEffectif Elastique Electrostatique /4 /Elment 9"Elongation Energie ;Entranement Equation Equilibre (Equiprojectif :*(Equiprojectivit :*(

Erreur PBEspace 'Exerc >Exponentiel +Expression 6"Extrieur B

FFigure BFluide &D

Fond ;Fondamental +Force 6;Frottement 5(

GGalilen 8Gaz 8Gomtrique %Giration (Glissement >:Gradient ?(Grandeur Gravitation Gravitationnel *(

HHarmonique (hlicodal

Horaire Horizontal +

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77

Hydraulique D

IIncertitude A(=Inclin DIncompressible ("Inertie "Initial D(Instantan Q4 0Intensit 63Intrieur BIsol(e) )6(

K

Kilogramme 8L

Liaison 4 liquide DLoi ;Longueur Lumineuse D'

MManomtre (Masse (Matriel *Matire 6Matrice 29Maximal 0"+Mcanique 5Mesure !;Mtre (Mobile 5(Module !4 634 Mole Moment "Mou Mouvement Moyen (Multiple I"'

N

Normale 0Notion /2

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78

Nutation + S(

OOprateur Ordonne A((Oscillatoire *(%

PParallle *Paroi Particule particulier 7BPermittivit $2Perpendiculaire

*"4 (Pesanteur $Phase 294 Plan (Plane V29Plaque 29Plein 9Point Polaire ;

Position &;4&'Potentiel Pousse Prcession 6+IPresse !Pression G'Principal DPrincipe +Produit

Projectile 2$;Propre ($Proprit 9BPseudo force 6;3Puissance "(Pulsation K

QQuadratique (Quantit

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79

RRayon ;I9Raction 2Rectangle (Rectiligne (Rduction (BRfrentiel &Relatif Relation ;1"remarque (Repre repos Rsultante 9Retard Y(Rotation Rotationnel (Roulement 4 ?(

SSatellite "9;Scalaire

Secondaire *)Seconde )Simple Sinus ASinusodal Solide A9Somme FSous multiple Sphre 6

Stabilit (Statique ("Superpos A(Surface VSymtrie 0(Systme 0

TTangente 0Temprature 6

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80

Temps Terrestre '+Thorme 0Tonneau Torseur /(0+(Trajectoire Translation ATravail "Tube A+

UUniforme 0(Unitaire 6

Unit 6Universel

VVariable G(vari G(Vase "Vecteur (4 F3Vertical ;3Vitesse arolaire V"

Volume Volumique

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Alphabet grec 8.


81

=>9Alphabet grec

[:;Prononciation

7_d[WMajuscule

7_7miniscule

Alpha WBta B Vgamma Udelta epsilon Ydzta ta thta Ziota kappa lambda mu nu xi omga H omicron

pi rh sigma tau upsilon

phi Khi u

psi

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Contenu de la deuxime partie !#-3'


82

yT:;mn:;]TXhg:XYTZ:;W[\:;X7g:; eZd:; WT[W

1 .V'.........................................................................

1 1.=9:.....................................................1 2."............................................................3 3.SE88D....................................................3 0%1.5......................................................................4 4.8D 12*................................................5 5.&-0"*-D"....................................6 0%2.5......................................................................6 0%3.5......................................................................6 6.1#-8D.......................................................8 0%4.5......................................................................8 0%5.5......................................................................10

7.9;SD:?SD...........................................11 8.

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Contenu de la deuxime partie !#-3'


83

23 1.+9; 0......................................................25 0%1.6......................................................................26 0%2.6......................................................................26 0%3.6......................................................................26 2."D9..........................................................27 0%4.6......................................................................28 3.88:3"S...................................29 0%5.6......................................................................

29 4.D9..........................................................32 5.2LD98D?93"0"+....34 0%6.6......................................................................36 6.D9.......................................................39 0%7.6......................................................................39 7.!1>......................................................40 0%8.6......................................................................41 0%9.6......................................................................42 8.D9!"D..........................................44 9.3"=S....................................................

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CENTRE UNIVERSITAIRE INSTITUT DES SCIENCESDE BECHAR EXACTES

DEPARTEMENT TRONC COMMUN

Licence, Master, DoctoratScience de la matire et sciences technologiques

(LMD/SM_ST)

COURS

DEUXIEME EDITION

LMD / PHYSIQUE-1

MECANIQUE DU POINT MATERIEL

Premire partie :

Cinmatique du point matriel

Documents

Mecpoint_1