CH1-4A

5/26/2018 CH1-4A

1/15

CMBUk 1 suIenma:TiccMnucrUbFatuChapitre I : Cinmatique du point matriel

g2

2sinv

g

cossinv

g

vcostgx

2

0

2

0

2

0

2

A

=

=

=

x;0cosv

g

dx

)x(zdxcosv

gtgdx

)x(dz22

0

2

2

22

0

3 kaMkMe)ag&n(n+gen!"tgcMnuc)am*

kg"bB.nDkUGredaen .renet e*Igman /

)t(v

dt

d

/

)t(v

dt

d tn

t ==

t)t(v)t(v =

#dl / = /(t) 4

CakaMkMe)ag&n(n+gen!"tgcMnuc Mt

)am*m:Eag$i?eTt e*Igman

zz*txx*ttz0x0yx ;)gtsinv(cosv)t(z)t(x)t(v +=+=+=

zz*txx*tz0x0 )t(v)t(v)gtsinv(cosv +=+

+

=

=

+

=

=

2

0

22

0

00z*t

2

0

22

0

00

x*t

)gtsinv(cosv

gtsinv

)t(v

gtsinv

)gtsinv(cosv

cosv

)t(v

cosv

[ ]2$

2

0

22

0

00

2

0

22

0

0x*t

)gtsinv(cosv

)gtsinv(cosgv

)gtsinv(cosv

cosv

dt

d

dt

d

+

=

+

=

[ ]2$

2

0

22

0

22

0

20

220

0z*t

)gtsinv(cosv

cosgv

)gtsinv(cosv

gtsinv

dt

d

dt

d

+

=

+

=

[ ]$20

22

0

44

0

22

0

22

0

22

z*t

2

x*t

2

t

)gtsinv(cosv

cosvg)gtsinv(cosvg

dt

d

dt

d

dt

d

+

+=

+

=

[ ] [ ]220

22

0

22

0

2

$2

0

22

0

2

0

22

022

0

2

2

t

)gtsinv(cosv

cosvg

)gtsinv(cosv

)gtsinv(cosvcosvg

dt

d

+

=

+

+=

+

==+

= cosgv

)t(v)gtsinv(cosv)t(/

)t(/

)t(v

)gtsinv(cosv

cosgv

dt

d

0

2

0

22

0

20

220

0t

CHAU SarwaddyMaster o !hysicsMaster o Micro"#ave

40

5/26/2018 CH1-4A

5/15


[ ]

+=

cosgv

)gtsinv(cosv)t(/

0

2

$2

0

22

0

>

enARtg;cMnuccab;epImecalGgFatuKWenA

Rtg;Kl; nRbBntM!uyeR"I#e!I# $

en!"tgcMnuc enH"tU$(an%g2)edIm t = t0= 0 ' etuenHkaMkMe)ag /

&n(n+gen!"tgcMnuc sM#dgeda*

[ ]

=

+====

cosg

v

cosgv

sinvcosv)0tt(//

2

0

0

2

$22

0

22

0

=

cosg

v/

2

0

> enARtg;cMnuc%bMputnKngen!"tgcMnuc27sbM8ut&n(n+g(9"tU$(a

n%g2) t 'tA ' gsinv 0 ' dUecHkaMkMe)ag /Aen!"tgcMnucenH

g

cosv

cosgv

cosv

cosgv

g

sinvgsinvcosv

)tt(//22

0

0

$$0

0

2

$2

00

22

0

AA=

=

+

===

g

cosv/

22

0A

=

> enARtg;cMnucFak;dl;dI


4&

5/26/2018 CH1-4A

6/15


en!"tgcMnucF+akdldI "tU$n%g2)

g

sinv2tt 0

== ' kaMkMe)ag /

&n(n+gen!"tgcMnucenH(9

=

=

+

===cosg

v

cosgv

v

cosgv

g

sinv2gsinvcosv

)tt(//20

0

$0

0

2

$2

00

22

0

==

cosg

v//

20

el56nF+akdldI

cMeBaHcMnuc)am*&n(n+grbscMnuc Ael56n2)rbs$asM#dgeda*

2

0

22

0

22 )gtsinv(cosv)t(z)t(x)t(v +=+=

en!"tgcMnucF+akdldI

2)#dl"tU$(aenaH(9g

sinv2t 0

=

eI*el56nrbs$aen!"tgcMnuc #dlCacMnucF+akdldI

0

22

0

22

0

2

0

0

22

0 vsinvcosvg

sinv2gsinvcosv)t(vv =+=

+==

0 vv =

e*Ige>I?@a el56ndldI es

5/26/2018 CH1-4A

7/15


Ca$iukT.rk ta&nG.k/ x nig y eI* * 4

TMMe@r 'en!2)edImcMnucrUbFatus;iten!"tgcMnuc

m*#dlmankUGredaen x0' 0; y0' 0 '

rk

k3 smIkar(n+grbscMnucrUbFatu y ' (x)23 kaMkMe)ag&n(n+gCaGnu(mn&n x

/ ' /(x)

dMe)aH"sa*

k3 kMntsmIkar(n+grbscMnucrUbFatu )x(yy=

e*Igman xv;vvvxv

yxyyxxyx

==+=+=

( ) ,Ctex2

xyxdxdx

xdxdyxdx

dy

2 +

=

=

=

=

4 en!2) 0t= cMnucrUbFatus;iten!"tgcMnuc

(x0' 0* y0' 0) ( ) Cte0y0xxy10x 000 ======

dUecHsmIkar(n+grbscMnucrUbFatu(9

2x

2

y(x) =

23 kaMkMe)ag&n(n+g )x(//=

kg"bB.nDkUGredaen.renet e*Igman CHAU SarwaddyMaster o !hysicsMaster o Micro"#ave

4$

5/26/2018 CH1-4A

8/15


n

2

n /

va

= eI* nt

/

)t(v

dt

d =

2y*t2x*t222

t

2222222

y

2

x

nt

dt

d

dt

d

x

dt

d

x/

/

x

/

vv

/

v

dt

d

+

+=

+=

+=

+==

tamTMnakTMngTI2 .renet e*Igman

yyxxt vv)t(vv +==

( )222

222222

xx*t

xdt

d

xxdt

d

v

v

dt

d

dt

d

++

=

+

=

=

( ) ( )2$

222

22

222

2

222

x*t

x

x

x

dt

dxx2

2

&

xdt

d

+

=+

+=

( )

222

2

&2222222

222

yy*t

x

xdt

dxxxx

dt

dx

x

x

dt

d

v

v

dt

d

dt

d

+

+

+

=

+

=

=

( )

( )( ) ( )2

$222

$

x

2

$222

222222

2

$222

xy*t

x

v

x

xx

x

v

dt

d

+

=

+

=+

+

=

dUecH ( ) ( ) 2222

$222

2

$222

244

t

xxx

x

dt

d

+

=+

+

+

=

kaMkMenag&n(n+g sM#dgeda*ken/am

( )

+=

++

==2

2

$222

222

2

222

t

x

x

x

dt

d

v/

23

2

x

1

R

+=

I-6-2/.

cMnucrUbFatum*eFIclnaes

5/26/2018 CH1-4A

9/15


v tamb+g&n(n+g y(x) 'rksMTuHrbscMnucrUbFatu nig kaMkMenag&n(n+gen!"tgcMnuc x = 0ebIsinCa(n+grbs$a

k3 ,a:ra:bUl y 'x2 * = Cte, ; 23 eGlIbCte**&

yx22

==

+

(3 y(x) ' x3nx en!"tg x ' & >3 y(x) 'sinxen!"tg 2x

= #dl , "

CaTMMe@reda*BiJak/aeT!

elIJaB(t(UbKJaB(t

ess&nTMM '

dMe)aH"sa*

3sMTuH&nclnarbscMnucrUbFatunigkaMkMe

)ag&n(n+g

e*Igrkken/amsMTuH )x(aa = nigkaMkMenag

)x(//= ' e*Igman n

2

tnnttnt /

v

dt

dvaaaaa

+=+=+=

#dl tt dtdv

a = 4 Ca$uikT.rsMTuH8Mb:H nig

n

2

n /

va

= 4 Ca$uikT.rsMTuH8M#kg

eda*cMnucrUbFatueFIclnaes

5/26/2018 CH1-4A

10/15


v

v

dt

d

dt

d*

dt

d

dt

dva

dt

dva

dt

dva

/

v

dt

d

/

va

0a0dt

dva

xx*t

2

y*t

2

x*t

tt

nt

n

2

tt

=+=

==

==

===

va

v

v

dt

d

dt

d yyy*t =

=

k3 ,a:ra:bUl ,Cte*xy 2 ==y 'x2 dy ' 2xdx * xv2vdtdxx2dtdy xy ==

( ) ( )x2xxyy*t

xavv

2xv2

dt

d

v

&

v

v

dt

d

dt

d+==

=

( ) ( )2x2

x

22

x

2

x

2

x2

2

2

2

x xav4axavv

4

v

ava ++=+

+=

( )222x2

y

2

x

2

xy x4&vvvvxv2v +=+==

( ) 0dv,Ctev*0dxxv+x4&dv,v2vdv2 2x222xx ===++=

( ) ( )xax4&

xv4a*

dt

dx,xv4

dt

dvx4& x22

2

x

2

xx

2x22 =+

==+

( )222

x

2

xx*t

x4&v

xv4

v

a

dt

d

+

==

( ) ( )222

x

22

2

x

22

2

xx

2

x

y*t

x4&v

v2

x4&

vx4v

v

2xav

v

2

dt

d

+

=

+

=+

=

( ) ( ) ( )

22

2

x

22

22

2

x

2222

4

x

2

2222

4

x

242

y*t

2

x*t

x4&

v2)x(a

&x4x4&

v2

x4&v

v4

x4&v

vx&v

dt

d

dt

dva

+

=

++

=+

+

+

=

+

=

mE:ageTt

( ) 2

x

222

22

2

x

2

n

v2

x4&vx/

x4&

v2

/

v)x(a)x(a

+=

+

===


4

5/26/2018 CH1-4A

11/15


( ) ( )

( )

2

x

2

yxy

2

x

2

0

2

x0

vv*00xvxv2v

v2

v0x//*v20xaa10x

====

=======

== 2&

/;2a 00

23 eGlIb &yx22

=

+

0ydy2xdx2&yx

,Cte*,Cte*&yx

222

2

2

2

22

=+=+

===

+

dt

dxx

dt

dyy0

ydyxdx2222

=

=

+

y

xvvv

xv

y x2

2

yx2y2

=

=

#t ( )2222

2

222

2

2

2

2

xx

&y&yx

=

==

+

22 xy

=

dUecH 22 x2222

y

x

xv

x

xv

=

=

( ) ,Cte

x

x&v

x

vxvvvv

222

222

x22

2

x

2

2

22

x

2

y

2

x

2 =

+=

+=+=

( )

( ) ( )

( )

0

x

dxx2xxdx2xv

xd

x&dvv2

2224

22222222

x222

22

xx =

+

+

( ) ( ) ( ) 0dxxx

x

xv

x

x&dv

222

2224

x

22

222

22

x =+

+

+

( ) ( ) ( ) 2222

x

2

x222

x

2

222

22

x

x

xvadx

x

xv

x

x&dv

=

=

+

e"BaH dtadv xx =

( ) 2222

xxx*t

xv

xv

v

a

dt

d

==


4%

5/26/2018 CH1-4A

12/15


=

=

=

22

x

22

xyy*t

x

xv

dt

d

vx

xv

dt

d

v

&

v

v

dt

d

dt

d

( ) ( )

+

=

22

x

22

2

&22

x

22

x

x

y*t

x

xa

x

xxvx22

&xv

vvdt

d

( ) ( )=

++

=

22

2

&222

x

2

x

2

x

22y*t

x

xvxxavx

vdt

d

( )( )

( ) ( )( )[ ]22

x

2

x

2

2

$22

2

$22

2

x

2

x

2

x

22

xxav

xvx

vxxavx

v+

=

++

=

dUecH2

y*t

2

x*t

dt

d

dt

dva

+

= manken/am

( ) ( )

( )[ ]222x

2

x

2

$2222

2

4222

4

x

22

xxav

xvxv

vxva +

+

=

( ) ( )[ ]222

x

2

x

2

2

224

x

2

222xxav

xvx

xa +

+

=

3en!"tg 0x=

( )( )

( )

0

x

x&x

xv0xa

vv0vx

xv

222

22222

2

x

2

x

2

x

2

x22

y

=

+

==

==

=

( )2

2

2

2

x

+

4

x

4

0

vvv0xaa

=

=

===

eI* ( ) 22

0

2

non0

v

/

va0xaa

=====

=2

0/


4+

5/26/2018 CH1-4A

13/15


=

=2

02

2

0 /*v

a

(3 #2/ekagmansmIkar ( ) x3nxxy = en!"tg 0x=

( )

( ) ( )xy vx3n&v*

dt

dxx3n&

dt

dy

dxx3n&dxx3n,dxdyx3nxy

+=+=

+=+==

( )

( )

4

v

2

v&xa

2

vvv2vv&xv

22

x

x

22

x

2

x

2

xy

===

====

( ) ( ) ( ) ( )( )[ ]2

2

xxxy

x3n&&x

x3n&vxavx3n&xv

++

+=+=

2224

&2

v

4

v

&

va

++=

m22//

v

22

v

&

v

&

va &

&

2244

& ===+=

m22/*22

va &

2

& ==

>3 ( ) xsinxy = en!"tg2

x =

( )

,vxcosv*dt

dxxcos

dt

dy

,

dxxcosdyxsinxy

xy ==

==

,xvcosvvvv 2x2222

x

2

y

2

x

2 +=+=

( )xcos&vv 2222x2 +=

( ) ,Ctev*0xdxsinxcosv2xcos&dvv2vdv2 $22x222

xx ==+=

( ) ,xdxsinxcos,vxcos&dv0 $2x222x +=

xcos&v

xsinxcosadt

dv222

x$2

xx

+==


4-

5/26/2018 CH1-4A

14/15


( )xcos&vv,xsinxcos

v

a

v

v

dt

d

dt

d222

x

$2

xxx*t

+

==

=

( ) [ ]xcos,av,xsinv

v,xcosdt

d

v

&

v

v

dt

d

dt

dx

2xx

yy*t +

==

=

( ) 2sin

2cos

2cos&v

v

xcos&v

v,xsinxcosdt

d0

222

x

$2

2x

222

x

$2

2x

x*t

+

=

+

=

=

=

+

=

=

2cos,av,

2sin

vdt

d

x

2

x

2x

y*t

cMeBaH ' 25 6 & #dl 5 ' 0* &* 2* , , ,

( ) 0&522cos2cos =+

=

; ( ) ( )5

&&522sin2sin =+

=

( ) ( )

( )0

&522

cos&

&522

sin&522

cos

v

v

dt

d

222

x

$2

2x

x*t =++

++

==

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) 0v&522

cos2

xvv,xcosv

&&*vv&v

v2

xa

dtd

dtdv

2xa

dtd

dtdva

v&v

&522

cosav&522

sinvdt

d

xyxy

&522

x

24

x

&52

2

42

2

2x

y*t

2

2x

x*t

2

y*t

2

x*t

2

x

&52

x

2

x

2x

y*t

=+

=

==

==

=

=

+

=

=

+

=

=

+

++

=

++

=

=

+

=

dUecH 22xx vvvv == ;

( )&52

22

av2xa +==

=


0

5/26/2018 CH1-4A

15/15


mE:ageTt


&

CH1-4A

Documents

Transcript of CH1-4A