Mécanique du point : référentiels non Galiléens (PCSI)

M m
L ω
t= 0 θ= 0
r0
r
L
θ=π
4
1er 42eme a0m.s−2
−a0m.s−2m
l m
a
a
a

a−→ Ω = ω−→ ez
RG= (−→ ex,−→ ey,−→ ez)RNG = (−→ e0
x,−→
e0
y,−→
ez)
θ RP=
(−→ er,−→ eθ,−→ e0
y)RNG
θ θ
RNG Ω−→ uz=−→ u0
zRG
−→
P=m−→ g=mg (cosθ−→ ur−sinθ−→ uθ)
−→
R=R−→ ur−→
Fie =mRsinθΩ2−→ u0
y=mRsinθΩ2(sinθ−→
ur+cosθ−→ uθ)
−→
uθ
0 = −mgsinθ +mRsinθΩ2cosθ
0 = cosθ −g
RΩ2RΩ2sinθ
θ= 0 θ=π cosθ =g
Ω2R
g
Ω2R<1
Ep(θ)−mgsinθ +mRsinθΩ2cosθ =Fθ=−1
R
∂Ep
∂θ
∂Ep
∂θ = 0 ∂2Ep ∂θ2>0
∂2Ep
∂θ2<0

∂2Ep
∂θ2=mgRcosθ −mR2Ω2cos2θ−sin2θ
θ= 0 ∂2Ep
∂θ2
0=mgR −mR2Ω2=mR g
RΩ2−1
g
RΩ2>1 Ω <pg R
θ=π∂2Ep
∂θ2
0=−mgR −mR2Ω2=−mR g
RΩ2+ 1<0
cosθ =g
Ω2R
sin2θ= 1 −cos2θ= 1 −g2
Ω4R2
∂2Ep
∂θ2
0
=mgR g
Ω2R−mR2Ω2g2 Ω4R2−1−g2 Ω4R2
=mg2
Ω2−2g2 Ω2−R2Ω2
=mR2Ω21−g2
R2Ω4
g
RΩ2<1 Ω >pg R
Ω<pg
RΩ>pg R
θ= 0
θ=π
cosθ =g
Ω2R
a
−→
Ω = ω−→ ezRG= (−→ ex,−→ ey,−→ ez)RNG = (−→ e0
x,−→
e0
y,−→
ez)
θ RP=
(−→ er,−→ eθ,−→ e0
y)RNG
θ θ
M m
z=ecos 2π
Lxv x
v
z
P=−mg
R
Fie =−m¨z=mv2e4π2
L2cos (vt/L)
v
0 = R−mg +mv2e4π2
L2cos 2π LvtR=mg +mv2e4π2 L2cos 2π Lvt
R Rmin =mg −mv2e4π2
L2v=vc=L 2πpg e

RT= (−→ ex,−→ ey,−→ ez)λ
RG= (−→ e0
x,−→
e0
y,−→
e0
z)
M m v0
−→
e0
z
RT= (−→ ex,−→ ey,−→ ez)λ
RG= (−→ e0
x,−→
e0
y,−→
e0
z)
M l m h θ0
(−→ ex,−→ ez)
x, y z
u=x+jy j2=−1u