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8/9/2019 INSA Toulouse 1A Mecanique Du Point Fascicule de TD
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ere
nd
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M
r = 10 cm
O
(t) OM
M
s1
{O, ex, ey} OM r (t)
OM
v
r
(t)
a
r
(t)
{O, e, e}
M
x = b.. cos
y = b.. sin b = Cte = { Ox, OM} = d
dt= Cte
t = 0
= 0
OM v a {O, e, e}
eT eN
eT =1
1 + (t)2
. e +t
1 + (t)2
. e eN =t
1 + (t)2
. e +1
1 + (t)2
. e
a eT eN aT aN a = aT. eT + aN. eN
aT aN eT eN
R
8/9/2019 INSA Toulouse 1A Mecanique Du Point Fascicule de TD
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{xOy}
I
r
J
K
O
{O, ex, ey} M t v M M K
a
M
{O, ex, ey}
{O, ex, ey}
M
R2 =
O
O
{O, ex, ey, ez}
O
R1 O
{O, ex, ey, ez}
O
O
=
O
O
M
OM(/) v(M/) a(M/) M
{O, ex, ey, ez} v(M/)
{O, ex, ey, ez} a(M/)
{O, ex, ey, ez}
OM {O, ex, ey, ez}
OM
v(M/) a(M/)
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M
M
{e, e}
||a||
||a|| = v dvd
e e
v
v0
R
v0
= = Cte
v
g
=
M
OM = r
R
Rv F
t = 0
OM0 = r0 = 0 O
{O, ex, ey} M
M
M
ar M ae M
ac M F(t)
M va aa r
ex ey
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v 300Km.h1
400
o
M
M
u
M
0 t = 0
= 0
v = v0
m M
F = k2 . e
m.2 ddt = C C
= 1 v = Cm dd . e + Cm ..e
a
a = Cm
2.2.
d2d2
+ . e
() (=0) = 0 dd (=0)
= 0
() = ...
8/9/2019 INSA Toulouse 1A Mecanique Du Point Fascicule de TD
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h
A
= a
z = h.
H = 2h
(z = 0)
z
g = GM(R+r)2
. er = g0 R2(R+z)2 . er g0 G M R r vlib(r)
U = 0
m
r0
vlib(r)
G = 6.671011m.s2
Mterre = 61024 kg Mlune = 7.31022 kg Rterre = 6400 km Rlune = 1700 km
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8/9/2019 INSA Toulouse 1A Mecanique Du Point Fascicule de TD
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8/9/2019 INSA Toulouse 1A Mecanique Du Point Fascicule de TD
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8/9/2019 INSA Toulouse 1A Mecanique Du Point Fascicule de TD
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ere
nd
a
{O, ex, ey, ez } {O, ex, ey, ez}
OO = a. ex + b. ey + c. ez
OM = x. ex + y. ey + z
. ez
OM = OO + OM
v(M/) = v(/) + v(M/)
v(M/) v(/)
= . ez = Cte
OM = A.t ex A
a(/) ac
(t) = 0
1 + cos (t)
0
OM
v
0
t
d
0 d
d
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ere
nd
m
M
0
{O, e, e, ez}
M O OM = R2
O
R1 O
{O, ex, ey , ez}
{O, ex, ey , ez}
= .t = ( ex, OO) = (ex, O
M)
ex ey
{O, ex, ey}
OM(/) v(M/) a(M/)
{O, ex, ey, ez}
e
v(M/) {O, ex, ey, ez}
a(M/)
{O, ex, ey, ez}
OM {O, ex, ey, ez}
OM
v(M/)
a(M/)
8/9/2019 INSA Toulouse 1A Mecanique Du Point Fascicule de TD
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B2 m k
f
B1
m
vi1 B2
B1 B2
xM
B2
12
k.x2M f.xM 12m.(vi1)2 = 0
R = 5 cm
2H = 20 cm
= 1, 6
= 1000 Kg/m3
G
m = 31.4 Kg
z0 = 5 cm
G
t = 0
G
A = . (H
z) . ez
z(t)
8/9/2019 INSA Toulouse 1A Mecanique Du Point Fascicule de TD
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8/9/2019 INSA Toulouse 1A Mecanique Du Point Fascicule de TD
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ere
nd
a
O, ex, ey, ez
() = A.e. cos
A
[0, ] = .t
e, e
ex, ey
OM
O, e, e
v
O, e, e
8/9/2019 INSA Toulouse 1A Mecanique Du Point Fascicule de TD
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a
O, e, e
eT, eN
[0, ]
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ere
nd
a
a ab cos sin tan
m1 m2
v1 v2
v1 v2 v1 v2
v
1
v
2
v1
v2
v
1
v
2
v1 =
2v2 + (k 1).v1k + 1
v2 =
2.k.v1 + (1 k).v2k + 1
k = m1/m2
m1 m2 v1 v2
v1 = v1.ex v2 = 0 m1 v2 m1 = m2
v1 = v1.ex v2 = 0 m1 >> m2
tL
t 0 < t < tL
t > tL g = 10m.s2
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{O,Ox,Oy} {ex, ey, ez} v = v0.ex {O, Ox, Oy}
m = 160g
d
dt (t) = .t
= 10 rad.s2 {e, e, ez} O
M
t = 0
= 0 tL
L = 225o
= 0.6m L = 1.5 m
vL [Ox)
t = tL
tL L
tL
ex ey {e, e, ez}
OM
v
a
{e, e, ez}
{ex ey, ez}
tL
v0
= 7 m.s
1
t = tL
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tL = 19.3o
y0 = 1.92 m vL = 11.4m.s1
fair = .v = 0.05 tL
ex
ey
vx(t) = vL. cos .exp
.tm
vy(t) =
vL. sin +
g.m
.exp
.tm
g.m
x(t) y(t)
vf
vf = 36.42 km.h1
1