[PDF] Correction du TD 1 Analyse dimensionelle et ordres de grandeurs

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15?→L

2→L2

mv2 ???????=MLT-2????F=ma ???=LT-2????F=P=mg ???=M-1L3T-2????F=Gm2r 2

T=LαMβ(L.T-2)γ

β= 0, α+γ= 0,-2γ= 1

β= 0, γ=-12

, α=12 ?l g ???? ???? ???? ???? ???? ??? ???????⎷2???? ???? ??????? ??⎷n???? ???? ?????? ???? ?? ???????n???? ???? ????? 2? ?? l? ????1,26l? ?? ????? ?? ??????? ??? ???? ???? ???? ???? 34
74
15 ?0,25?3? ?? ? ????Vcell=Vhumain.2-40?0,25.(103)-4?

0,25.10-12?3?

????Lcell=V13 10 -4?? ??????? ?? ???? ??????? ?Scheveux=? ?? ? ??2?10-2m2 ??????? ??????? ??? ??? ??????? ?Scell=L2cell?10-8m2?

ScheveuxS

cell?106??????? ??? ?? ????? ddx •(xn)?=nxn-1 •(1x n)?=-1x n)?= (1f(x))?=-f?(x)f 2(x) ??(xα)?=αxα-1? •(⎷x)?=12 ⎷x (x1/2)2=x= (⎷x)2 •(ln(4x+ 3))?=44x+3 •(exp(-2x+ 3))?=-2exp(-2x+ 3) •(cos(4x-2))?=-4sin(4x-2) h→0? ?????Δx? f ?(x) =dfdx (sin(x))?= limh→0sin(x+h)-sin(x)h = lim h→0sin(x)(cos(h)-1) + sin(h)cos(x)h = lim h→00 + cos(x)sin(h)h = cos(x) (cos(x))?= limh→0cos(x+h)-cos(x)h = lim h→0cos(x)(cos(h)-1)-sin(x)sin(h)h = lim h→00-sin(x)sin(h)h =-sin(x) = 1??limh→0cos(h) = 1? ddx dfdx =dexdx =ex????df=exdx? dgdt =det+φdt =et+φ????dg=et+φdt? dhdt =dπ3φ+4tdt =d(3φ+4t)dt ?(g◦f)?(x) =f?(x)(g?◦f)(x) dkdx =d(xyt+φ)dx = (yt+φ)xyt+φ-1??dkdt =d(xyt+φ)dt =d(yt+φ)dt d(xyt+φ)d(yt+φ)= yx yt+φ? ????dk= (yt+φ)xyt+φ-1dx+yxyt+φdt? adf=?b adfdx ?5

0x2dx= [x33

]50= 125?

0sin(ωt)dt= [-cos(ωt)ω

]π0=1ω (1-cos(ωπ))? ?0 -32 ?(2x+ 3)dx=?0 -32 (2x+ 3)1/2dx= [23 (2x+ 3)3/2]0-32 = (-3)3/2?

0xln(x)dy=xln(x)?

100

0dy=xln(x)[y]1000= 100xln(x)?

????e e 0

1ln(S)dS= [Sln(S)]e0-?

e 0 S1S dS= [Sln(S)-S]e0= 0 ???limS→0Sln(S) = 0? ?P=m?g???? ???? ?? ?????F=m?a?? ??a=?g? ????? ?? ????? ??? ??? ?????z? ?? ???? ????? ??????az=-g??ax= 0? dv xdt =ax??dvzdt =az? ????vx=v0x= 0??vz=-gt+v0z=-gt? dxdt =vx??dzdt =vz? ????x=x0= 0??z=-12 gt2+z0=-12 gt2+l? z= 0? -12 gt2z=0+l= 0?t2z=0=2lg t z=0=?2lg ????vz=0=-g?2lg =-⎷2gl ?P=m?g??????F=m?a→?a=?g? ????ay=-g??ax= 0? dvydt =ay????vy(t) =-gt+v0y? ??v0y=v0sin(α)???? v y(t) =-gt+v0sin(α)? dv xdt =ax????vx(t) =v0x? ??v0x=v0cos(α)????vx(t) =v0cos(α)? dydt =vy????y(t) =-12 gt2+v0sin(α)t+y0 dxdt =vx????x(t) =v0cos(α)t+x0=v0cos(α)t =vy= 0??? ? ???? ?-gtF+v0sin(α) = 0? ????tF=v0sin(α)g ?? ? ???? ???????yF=-12 gt2F+v0sin(α)tF+y0=-12gv20sin2(α) + 1g v20sin2(α) +y0=12gv20sin2(α) +y0? ????? ???x????? ??? ??????? ?? ??????? ?? ?????? ?? ???? ??????? y= 0? ???? ?-12 gt2+v0sin(α)t+y0= 0? ????? ?? ??????? ?? ?????? t

P=v0sin(α) +?v

20sin2(α) + 2gy0g

x

P=x(tP) =v0cos(α)tP=v0cos(α)v0sin(α) +?v

20sin2(α) + 2gy0g

0sin(α)+⎷v

20sin2(α)g

=2v0sin(α)g x

P=2v20cos(α)sin(α)g

???? ?? ???????dα? dx

Pdα

=dv

0cos(α)v

0sin(α)+⎷v

20sin2(α)+2gy0g

dα dx

Pdα

=v20g (-sin2(α)+cos2(α))+v0g (-sin(α)?v

20sin2(α) + 2gy0+cos2(α)v20sin(α)?v

20sin2(α) + 2gy0)

dx

P=v20g

(cos(2α)) +v0sin(α)g (?v

20sin2(α) + 2gy0-v20cos2(α)⎷

v

20sin2(α)+2gy0dα

T=Mα(L.T-1)β(L.T-2)γ

α= 0, β+γ= 0, β+ 2γ=-1

α= 0, β= 1, γ=-1

???? ?? ?tver?vzg

T=MαLβ(L.T-1])γ

α= 0, β= 1, γ=-1

???? ?? ?thor?dv x? ???? ????? ?? ?????thor=tver? ?? ? ?????L?vxvzg

L?v20cos(α)sin(α)g

y MLT -2=MαLβ(LT-1)γ

α= 1γ= 2β=-1

????F?mv2l =12005023,6228 = 1,36.108?? ?v ref=rives=?vb+?vc? ?? ? ????vref=rives=?v ?v t ref=rives=lv ref=rivescos(α) t ref=rives=l?v

2c+v2bcos(α)

t ref=rives=l?v 2b=lv b t ref=courant=lv b ???? ?? ?????t?f(lv c,lv -1? ?? ??????? ?? ??????? ???? ??? ???-1? b) = 0,895???= 51,3

2b-v2c=⎷0.39 = 0,62???-1?

??t=lv ref=rives=620,62= 100? ???? ? ??? ??? ??????? ?????? ?? ?me= 9,11.10-31???mp= 1.67-27???qp=-qe=

1,60.10-19??14π?0= 8,98.10-9???-1?G= 6,67.10-11?2??-2?

q eqp4π?0r2 F elecF grav=14π?0q eqpGm emp?10-810-3810 -1010-3010-27?1019 (Flec+Fgrav)m e=(Flec(1+10-19))m e? ????? ??? ?? ??????F? [a] =LT-2 LT -2= (LT-1)αLβ ???? ??????? ??????? ??? ?????? ? ?? ??????F? ?????? ?????? ?? ?????? ?? ??? ?? ?(r+dr)2= (dl)2+r2? ????2rdr+ (dr)2=v2(dt)2? ?? ?? ???? ?F?? ??????vF=Fm ?F??? ??????? ?????? ???? ?? ?????? ????? ??? ??????? ????? ?? ?????? 1r

F2m(dt)2=v22r(dt)2? ????Fm

=v2r 1r a?Cter

T=longueur d?untourvitesse

=2π(RT+h)v ?? ?a=FGm S=v2R

T+h? ???? ?? ???? ??????v2=GMTR

T+h? ?? ??????? ??

?????? ?? ??????T=2π(RT+h)32⎷GM T? R

T+h=?T⎷GM

T2π?

23
h=?86400⎷4.10142π? 23
-RT= 4,229.107-0,64.107= 35890km ?? ?????? ??? ?? ???????MT= 6.1024???RT= 6400?? ??G=

6,67.10-11? ?2??-2?

t= 0? ?? ?????? ? ??? ????? ?θ1(0) = 0? ?? ?????? ? ?θ2(0) =π? ??? ?????? dθ1dt =dθ2dt =ω? ????θ1(t) =ωt???θ1(0) = 0??θ2(t) =ωt+π ???θ2(0) =π? ?? ?????? ????θ2(t) =θ1(t) +π? x

1=rcos(ωt)??x2=rcos(ωt+π)?

?? ???????θ1(t) =ωt?θ2(t) =ωt+π2 ? ??x1=rcos(ωt)??x2= rcos(ωt+π2

2???? ?? ??????

??d2x1dt dx1dt =-rωsin(ωt)??d2x1dt

2=-rω2cos(ωt)? ??

2d2x1dt

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