Problème 1 : Masse pesante suspendue à un ressort (10 points)

m
~
Fr=−kz~ez~ Fg=−mg~ezk > 0
g > 0
z v
¨z(t) + ω2z(t) = −g t ω
k m
z(t= 0) = 0
v(t= 0) = 0
T
z(t)zmin zmax
zmin zmax m g k
~
F=~ Fr+~ Fgzmin
zmax
V(z)~ F=~ Fr+~ Fg
V(z= 0) = 0
z0V(z)V0=
V(z=z0)V(z)z0
z v
t= 0 z(t= 0) = 0 v(t= 0) = 0
−→

MSG
−−→
OM =~r =r ~er
r ~r −→
F=−GMSm
r2~er
~
L
~r ~v ~ L
r θ
C=r2˙ θ
R
v0=||~v||
GMSv0R
r=p
1+ecos(θ)p≥0
rP
rP=R
2
2v0
~v ~a
~v = ˙r ~er+r˙ θ ~eθ
~a = (¨r−r˙ θ2)~er+ (r¨ θ+ 2 ˙r˙ θ)~eθ