INSA Toulouse 1A Mecanique Du Point Fascicule de TD

8/9/2019 INSA Toulouse 1A Mecanique Du Point Fascicule de TD

1/22

ere

nd


2/22

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


3/22

{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/)


4/22

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


5/22

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

() = ...


6/22

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


7/22


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14/22

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


15/22

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/)


16/22

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)


17/22


18/22

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


19/22

a

O, e, e

eT, eN

[0, ]


20/22

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


21/22

{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


22/22

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

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INSA Toulouse 1A Mecanique Du Point Fascicule de TD