Référentiels non galiléens Notes de cours

R
m~a/R(M) = X~ f→M
1 an
24 h
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R0R R0
R0R M R R0
•R
(O, ~ux, ~uy, ~uz)
•R0
(O0, ~ux0, ~uy0, ~uz0)
R0
R
~ux0=~ux
~uy0=~uy
~uz0=~uz
M
•−−→ OM (t) = x(t)~ux+y(t)~uy+z(t)~uz
R
•−−−→ O0M(t) = x0(t)~ux+y0(t)~uy+z0(t)~uz
R0
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M
•~v/R(M) = d−−→
OM(t)
dt /R
= ˙x(t)~ux+ ˙y(t)~uy+ ˙z(t)~uzR
•~v/R0(M) = d−−−→
O0M(t)
dt /R0
=˙ x0(t)~ux+˙ y0(t)~uy+˙ z0(t)~uzR0
~v/R(M) = d−−→
OO0(t)
dt !/R
+ d−−−→
O0M(t)
dt !/R
=~v/R(O0) + d(x0(t)~ux+y0(t)~uy+z0(t)~uz)
dt /R
~v/R(M) = ~v/R(O0) + ˙ x0(t)~ux+˙ y0(t)~uy+˙ z0(t)~uz
⇒
~v/R(M)M R ~v/R0(M)M
R0
~v/R(M) = ~v/R0(M) + ~ve
~veR0R
P M t R0
R0R
M t P
~ve=~v/R(P)
t0P M
P0
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M
•~a/R(M) = d~v/R(M)(t)
dt /R = ¨x(t)~ux+ ¨y(t)~uy+ ¨z(t)~uzR
•~a/R0(M) = d~v/R0(M)
dt /R0=¨
x0(t)~ux+¨ y0(t)~uy+¨ z0(t)~uzR0
~v/R(M) = ~v/R0(M) + ~ve⇒
~a/R(M) = d~v/R0(M)
dt /R
+d~ve
dt /R
= d˙
x0(t)~ux+˙ y0(t)~uy+˙ z0(t)~uz
dt !/R
+~ae
~a/R(M) = ¨ x0(t)~ux+¨ y0(t)~uy+¨ z0(t)~uz+~ae
⇒
~a/R(M)M R ~a/R0(M)M
R0
~a/R(M) = ~a/R0(M) + ~ae
~aeR0R
R
m~a/R(M) = X~ f→M=m~a/R0(M) + m~ae
⇒
R0~ F
~
fie =−m.~ae
R0R
~ve(M) = ~v0=−−→ cste
~v/R(M) = ~v/R0(M) + ~v0
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R0R
R0~ve=−−→ cste ⇒~ae=~ 0⇒
R0R
~ae=a0~ux~
fie =−m.~ae=−m.a0~ux=−−−→ grad (m.a0x)
⇒
~ae=a0~ux
Ep=m a0x
R0
R ~ve= +v(t)~ux~ae=−a ~ux
R0R0
R ~ve= +v0~uθ~ae=−v2
0
R0~ur
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