MECANIQUE RATIONNELLE

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UMBB Boumerdès, Faculté des sciences, Département de physique Cours exercices, Mécanique Rationnelle : TCT et LMD-ST sem :3 A.KADI L’accélération se déduit par dérivation : → → → 0 0 2 0 → → 0 d V ( O2 ) d V ( O2 ) 0 0 γ ( O2 ) = = + Ω 2 ∧V ( O2 ) avec : dt dt d 2 → 0 ( O2 V dt ) → = 0 car • ψ = Cte d’où : → 0 γ ( O ⎧ 0 ⎪ ) = ⎨ 0 ⎪ R ⎩ψ 2 0 ∧ • ⎧ ⎪2 0 ⎨ Lψ ⎪3 R ⎩ 0 0 • = ⎧ 2 ⎪ − Lψ ⎪ 3 ⎨ 0 ⎪ 0 ⎪ R ⎩ 0 • 2 → 4. Vitesse V 0 ( N) dans R , sachant que : V Nous avons par la cinématique du solide : V −−→ = MN ⎧ 0 ⎪ ⎨ 0 = ⎪ R ⎩− b R 3 2 ⎧ 0 ⎪ ⎨ bsinθ ⎪ ⎩− bcosθ 2 → 1 → 1 → → ( M ) = α( t) x2 + β ( t) y2 → 1 → 1 3 −−→ ( N) = V ( M ) + Ω ∧ MN → 1 V → ( N) = α ( t) x 2 → + β ( t) y 2 ⎧0 ⎪ + ⎨0∧ • ⎪ R ⎩θ R 2 2 ⎧ ⎪ ⎨− ⎪ ⎩ 0 bsinθ bcosθ = • ⎧ ⎪ α( t) + bθ sinθ ⎨β ( t) ⎪ 0 R ⎪⎩ 2 5. Moment cinétique de la tige AB au point O par rapport à et exprimé dans R ; R0 1 → → −→ → → 0 0 0 0 ( O) = σ ( C) + OC∧ m1 V ( C) = σ ( C) = I C σ . Ω car : V 0 ( C) = 0 et → 0 2 → → → 0 2 Ω → 0 1 = Ω ⎡ ⎤ ⎢0 0 0 ⎥ ⎡ ⎤ ⎡ ⎤ → ⎢ ⎥ 0 ⎢ 0 ⎥ 0 m L m L m L σ ( O) z → 2 2 • → 2 • → 0 1 ⎢ ⎥ ⎢ ⎥ 1 1 = I ⎢ ⎥ C . Ω1 = 0 0 ⎢ 0 = = = ⎢ ⎥ • ⎥ 0 ψ z2 ψ 1 3 ⎢ 2 • ⎥ 2 ⎢ ⎥ ⎢⎣ ⎥ m 3 3 1L m ⎦ ⎢ ⎥ 1L R ψ ψ 2 ⎢0 0 ⎥ R ⎣ 3 ⎦ 2 R ⎣ 3 ⎦ 2 6. Moment cinétique de la plaque au point O 2 par rapport à et exprimé dans R ; → 0 ( O2 ) = I O2 σ → 2 3 . Ω R2 2 353
UMBB Boumerdès, Faculté des sciences, Département de physique Cours exercices, Mécanique Rationnelle : TCT et LMD-ST sem :3 354 A.KADI b a m R R a m m b b a m R O 2 ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ + = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ + = • • → 0 0 ( 3 0 0 3 0 0 0 3 0 0 0 ( 3 ) ( 2) 2 2 2 2 2 2 2 2) 2 2 2 2 0 θ θ σ → • → + = 2 2 2 2 2 0 ) ( 3 ) ( z b a m O θ σ 8. Energie cinétique du système. ) ( ) ( ) ( 3 2 S E S E Système E C C C + = C T C S I C V m S E → → → Ω Ω + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 0 1 3 0 1 2 0 1 2 2 1 ). ( . 2 1 ) ( 2 1 ) ( • • • = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 2 2 1 2 2 1 2 1 2 2 6 0 0 3 0 0 0 3 0 0 0 0 ). (0,0, 2 1 ) ( ψ ψ ψ m L R m L m L R S E C C T C S I O V m S E → → → Ω Ω + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 0 3 3 0 3 2 2 0 2 3 2 1 ). ( . 2 1 ) ( 2 1 ) ( ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = • • • 0 0 3 0 0 0 3 0 0 0 ( 3 ,0,0) ( 2 1 3 2 2 1 ) ( 2 2 2 2 2 2) 2 2 2 2 3 θ θ ψ R a m m b b a m R L m S E 2 C • • • • + + = + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = 2 2 2 2 2 2 2 2 2 2 2 2 3 ) ( 6 9 2 ) ( 3 2 1 3 2 2 1 ) ( θ ψ θ ψ b a m L m b a m L m S E 2 C • • • + + + = 2 2 2 2 2 2 2 2 2 1 ) ( 6 9 2 6 ) ( θ ψ ψ b a m L m m L Système E C • • + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = 2 2 2 2 2 2 2 1 ) ( 6 9 2 6 ) ( θ ψ b a m L m m Système E C
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CLASSES PREPARATOIRES AUX GRANDES E
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Préface Quand Ali KADI m’a amica
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UMBB Boumerdès, Faculté des scien
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MECANIQUE RATIONNELLE

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