[PDF] Mécanique rationnelle Dans ce cours la mé

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~AB??????? ??? ???????A (A;B)"(C;D) ~AB=~CD? ?? ?????? ?? ???????d????? ?????? ????? ? ?? ?? ? ??????? ?? ???????~u? ?????? ? ?? ? ?? ? ??????? ? ?? ?? ~AB? ?????? ? ?? ? ?? ? ????? ? ?? ?? ~c=~a+~b = (ax~1x+ay~1y) + (bx~1x+by~1y) = (ax~1x+bx~1x) + (ay~1y+by~1y) = (ax+bx)~1x+ (ay+by)~1y ~c=~a+~b (cx;cy) = (ax;ay) + (bx;by) = (ax+bx;ay+by) ?? ?????~a+~b= (ax+bx;ay+by) = (bx+ax;by+ay) =~b+~a? ~a+(~b+~c) = (ax;ay)+(bx+cx;by+cy) = (ax+bx+cx;ay+by+cy) = (ax+bx;ay+by)+(cx;cy) = (~a+~b) +~c? ?? ??~a+~0 = (ax+ 0;ay+ 0) =~0 +~a=~a? ??????~a????? ?????? ????? ?? ??~a+~a0=~0,~a0=~a? ~AB+~BA=~AA=~0 ~AB=~BA ~c=~a~b = (ax~1x+ay~1y)(bx~1x+by~1y) = (ax~1xbx~1x) + (ay~1yby~1y) = (axbx)~1x+ (ayby)~1y ~c=~a~b (cx;cy) = (ax;ay)(bx;by) = (axbx;ayby) ?? ?????~a~b= (axbx;ayby) = (bx+ax;by+ay) =~b+~a=(~b~a)? ~AO??~OB? ???? ????? ????

AB=~AO+~OB

~OB+~AO ~OB~OA

OB~OA= (Bx~1x+By~1y)(Ax~1x+Ay~1y)

=Bx~1xAx~1x+By~1yAy~1y = (BxAx)~1x+ (ByAy)~1y ~AB? ?????? ? ??(BxAx)???? ?? ~v(1;4) +~u(5;1) =~w(6;5)????? ? ?1 + 5 = 6 ~w? ????? ? ?? ? ????~v?? ???? ?? ? ????~u????? ? ????? ? ??4 + 1 = 5?? ~v(2;3)~u(1;1) =~w(1;4)????? ? ?21 = 1 ? ??21 = 1??~w? ????? ? ?? ? ????~v?? ???? ? ????? ? ??(1) = 1????~u????? ? ????? ? ??

3(1) = 3 + 1 = 4??

k????? ?????? ????? ??? ?? ??????? ???? k~a ? ?? ?????? ???? ??????? ??? ??k= 0? ~a 0=k~a =k(ax~1x+ay~1y) ~a 0=k~a (a0x;a0y) =k(ax;ay) = (k ax;k ay) ?? ?????(k1+k2)~a= (k1+k2)(ax;ay) = ((k1+k2)ax;(k1+k2)ay) = (k1ax;k1ay)+(k1ax;k2ay) = k

1(ax;ay) +k2(ax;ay) =k1~a+k2~a?

?? ?????(k1k2)~a= (k1k2)(ax;ay) = (k1k2ax;k1k2ay) =k1(k2~a) =k2(k1~a) =k2k1~a? ?? ?????1~a=~a ?? ?????0~a=~0

3x+ 4y= 4 (2)

2x+y= 1 ( 3),(

6x+ 8y= 8

6x+3y=3,0x+ 5y= 5,y= 1(

3x+ 4y= 4 (1)

2x+y= 1 ( 4),(

3x+ 4y= 4

4x+4y=4, x+ 0y= 0,x= 0

~AB= (BxAx)~1x+ (ByAy)~1y? ?????? ????? ????? ??? ??? ????x??y????

B??? ?? ??????dy=Ay?

?????Ay=B0y??Bx=B0x? ?? ? jj ~ABjj2=jABj2 =jAB0j2+jB0Bj2 = (B0xAx)2+ (ByB0y)2 = (BxAx)2+ (ByAy)2 jj ~ABjj=q(BxAx)2+ (ByAy)2 (x;y)? ?? ?jj~ABjj=p(62)2+ (31)2=p20 jj~ajj=qa

2x+a2y

jj ~1rjj2= cos2+ sin2= 1) jj~1rjj= 1

1a=~ajj~ajj

jj ~1ajj=jj~ajjjj~ajj= 1

2x+a2y0?

jj~ajj=qa

2x+a2y+a2z

~a~b ap(x)q(x)dx? :EE!R: (x;y)!< xjy > ? ?????? ?< xjx >= 0)x= 0 ~a~b=axbx+ayby ~a~b=jj~ajj:jj~bjj:cos ~a~b=jj~a0jj:jj~bjj ~a~b=jj~ajj:jj~b0jj ?? ?????~a~a=axax+ayay=a2x+a2y0? ~0~0 = 0:0 + 0:0 = 0? ?? ?????~a~b=axbx+ayby=bxax+byay=~b~a? ?? ?????~a(~b+~c) =ax(bx+cx) +ay(by+cy) =axbx+ayby+axcx+aycy=~a~b+~a~c? ?? ?????~a~0 =~0~a=ax0 +ay0 = 0? ?? ?????~a(~b~c) =k~ak~a??(~a~b)~c=k0~ck~c? ?????~a(~b~c)6= (~a~b)~c ? ?? ?~AB~AC=jj~ABjj:jj~ACjj:cos3

5:3:12

= 7;5 ~a~b=jj~ajj:jj~bjj:cos2 =jj~ajj:jj~bjj:0 = 0 ~a~b=jj~ajj:jj~bjj:cos0 =jj~ajj:jj~bjj:1 =jj~ajj:jj~bjj ~a~b=jj~ajj:jj~bjj:cos=jj~ajj:jj~bjj:(1) =jj~ajj:jj~bjj ~a~b=axbx+ayby=qa

2x+a2yqb

2x+b2ycos

,cos=axbx+aybyq (a2x+a2y)(b2x+b2y)

2:5+3:2p((2)2+32)(52+22)=4p377

!= arccos4p377 102o
jj~a0jj=~a~1d=jj~ajj:cos a jjax~1xjj= (ax~1x+ay~1y)~1x =ax(~1x~1x) +ay(~1y~1x) =ax~1x2+ay~1x~1y =axjj~1xjj2 =ax jjay~1yjj= (ax~1x+ay~1y)~1y =ax(~1x~1x) +ay(~1y~1x) =ax~1x~1y+ay~1y2 =ayjj~1yjj2 ap(x)q(x)dx???? ??????? ??? ?????jjpjj=p< pjp >=qR b ap2(x)dx jj~ajj2=jj~ajj:jj~ajj:cos0 =~a~a jj~ajj=p~a ~a~b=axbx+ayby+azbz ~a?~b,~a~b= 0 jj~ajj2+jj~bjj2+ 2jj~ajj jj~bjjcos? ?? ?~a+~b(ax+bx;ay+by)? ????? jj~a+~bjj2= (~a+~b)(~a+~b) =~a~a+~a~b+~b~a+~b~b =~a2+ 2~a~b+~b2 =jj~ajj2+jj~bjj2+ 2jj~ajj jj~bjjcos

2jj~ABjj jj~ADjjcos120o= 52+ 32+ 5:3:(12

) = 26;5! jj~ACjj=p26;55;15cm? ?[~a;~b]???? ??? ???? ?? ?????? ~a~b=~

1x~1y~1za

xayazb xbybz= (aybzazby)~1x+ (azbxaxbz)~1y+ (axbyaybx)~1z ~a~b=~

1x~1y~1za

xayazb xbybz=~

1x~1y~1zb

xbybza xayaz=~b~a ~a(~b+~c) =~

1x~1y~1za

xayazb x+cxby+cybz+cz=~

1x~1y~1za

xayazb xbybz+~

1x~1y~1za

xayazc xcycz????? ~a(~b+~c) =~a~b+~a~c k(~a~b) =k~

1x~1y~1za

xayazb xbybz=~

1x~1y~1zka

xkaykazb xbybz=~

1x~1y~1za

xayazkb xkbykbz????? k(~a~b) =k~a~b=~ak~b ?? ?????~a~0 =~0~a=~

1x~1y~1za

xayaz0 0 0= ~0? ~a(~b~c) =~

1x~1y~1za

xayazb yczbzcybzcxbxczbxcybycx= (ay(bxcybycx)az(bzcxbxcz))~1x+::: = (aybxcyaybycxazbzcx+azbxcz))~1x+::: (~a~b)~c=~

1x~1y~1za

ybzazbyazbxaxbzaxbyaybxc xcycz= ((azbxaxbz)cz(axbyaybx)cy)~1x+::: = (azbxczaxbzczaxbycy+aybxcy)~1x+::: = (aybxcyaxbycyaxbzcz+azbxcz)~1x+::: ~a(~b~c)6= (~a~b)~c ~a~b=jj~ajj:jj~bjj:sin2 =jj~ajj:jj~bjj:1 =jj~ajj:jj~bjj ~b~a=jj~bjj:jj~ajj:sin32 =jj~bjj:jj~ajj:(1) =jj~ajj:jj~bjj ~a~b=jj~ajj:jj~bjj:sin0 =jj~ajj:jj~bjj:0 = 0 ~a~b=jj~ajj:jj~bjj:sin=jj~ajj:jj~bjj:0 = 0 sin=jj~a~bjjjj~ajj jj~bjj=s(aybzazby)2+ (azbxaxbz)2+ (axbyaybx)2(a2x+a2y+a2z)(b2x+b2y+b2z) ~ak~b,~a~b=~0 (~a~b)~c ~a(~b~c) = (~a~c)~b(~a~b)~c (~a~b)~c= (~a~c)~b(~a~c)~a jj~a~bjj2+ (~a~b)2=jj~ajj2jj~bjj2 jj~a~bjjjj~ajj jj~ajj! 2 ~a~bjj~ajj jj~bjj! 2 = 1 sin

2+ cos2= 1

jj~a~bjj2+ (~a~b)2= (a2x+a2y+a2z)(b2x+b2y+b2z) (a2+b2)(c2+d2) = (ac+bd)2+ (adbc)2 ~a(t) =_~v(t) =~r(t) ~r(t) =~R0R(t) =~OR(t)~OR(t0) =~OR(t)~OR0 ????? ???? ?????? ???ry=f(rx)? s 1s 2t1t 2,s1t 1s2t 2 ???vmoy=502 kmh = 25kmh ~v moy=~OR2~OR1t

2t1=~R1R2t

2t1=~Rt

~v moy(t1;t2) =~OR(t2)~OR(t1)t

2t1=~r(t2)~r(t1)t

2t1 ~v(t) = limt0!t~vmoy(t;t0) = limt0!t~r(t)~r(t0)tt0=d~rdt t? ~r+d~r~rt+dtt=d~rdt ~v(t) =d~r(t)dt =_~r(t) ?? ??????? ?t?? 1 ms

2?? ?m:s2?

~a=~vt ~a moy=~v2~v1t

2t1=~vt

~a moy(t1;t2) =~v(t2)~v(t1)t 2t1 ~a(t) = limt0!t~amoy(t;t0) = limt0!t~v(t)~v(t0)tt0=d~vdt =_~v(t) ~v+d~v~vt+dtt=d~vdt ~a(t) =d~v(t)dt =_~v(t) ???????t? ~a t=djj~vjjdt ~1t ~a n=jj~vjj2 ~1n ~1n(t)jt=t0?? ?? ??????? ~v(t) =_~r(t) ~a(t) =_~v(t) ~v(t) =d~r(t)dt ~a(t) =d~v(t)dt ~a(t) =d~v(t)dt =dd~r(t)dt dt =d2~r(t)dt

2=~r(t)

~v(t) =_~r(t) r(t) =ZZ r(t)dt2=ZZ a dt 2=ZZ g dt 2=Z Z 10dt dt=Z (10t+C1)dt= 10t2=2 +C1t+C2 a x= 0 a y= 0,( dv xdt = 0 dv ydt = 0,( v0x= 0 v

0y= 0,(

v v v x=v0x v y=v0y,( dr xdt =v0x dr ydt =v0y,( dr x=v0xdt dr y=v0xdt Rrx r

0xdrx=Rt

t

0v0xdtRry

r

0ydry=Rt

t

0v0ydt,([rx]rx

r

0x=v0x[t]t

t 0 [ry]ry r

0y=v0y[t]t

t 0,( r xr0x=v0x(tt0) r yr0y=v0y(tt0) r x(t) =v0x(tt0) +r0x r y(t) =v0y(tt0) +r0y,( rxr0xv

0x+t0=t

r yr0yv

0y+t0=t

rxr0xv

0x+t0=ryr0yv

0y+t0 ,ry=v0yv

0xrx+ (r0yr0xv0yv

0x) ?? ??????m=v0yv

0x??p=r0yr0xv

0yv ,ry=m rx+p a a dv xdt =ax dv ydt =ay,( dv x=axdt dv y=aydt

Rdvx=RaxdtRdvy=Raydt,(

v xv0x=ax(tt0) v yv0y=ay(tt0) v x=ax(tt0) +v0x v y=ay(tt0) +v0y,( dr xdt =ax(tt0) +v0x dr ydt =ay(tt0) +v0y,( dr x= (ax(tt0) +v0x)dt dr y= (ay(tt0) +v0y)dt Rrx r

0xdrx=Rt

t

0(ax(tt0) +v0x)dtRry

r

0ydry=Rt

t

0(ay(tt0) +v0y)dt,8

:[rx]rx r

0x=axh

t22 t0ti t t

0+v0x[t]t

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