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 3.2. Moment cinétique et moment dynamique en O → → −−→ → ( O1 2 1 1 0 0 0 σ O) = I . Ω + OO ∧ mV ( O ) or V → 0 ( O ) = 1 ⎧ 0 • ⎪ ⎨Lθ ⎪ R ⎩ 0 1 ⎡A → 0 σ ( O ) = ⎢ ⎢ 0 R ⎢⎣ 0 1 0 C 0 • ⎛ ⎞ 0⎤ ⎜ϕ ⎟ ⎛ L⎞ ⎜ ⎟ 0 ⎥ ⎜ ⎟ ⎥ 0 + ⎜ 0 ⎟ ∧ m ⎜ • ⎟ C⎥ ⎜ ⎟ ⎜ ⎟ ⎦ θ ⎝ 0 ⎠ ⎝ ⎠ R1 R 1 ⎧ 0 ⎪ ⎨Lθ ⎪ R ⎩ 0 1 • = • ⎛ Aϕ ⎜ ⎜ ⎜ ⎜ R ⎝ 1 0 ⎞ • ( + ) ⎟ ⎟⎟⎟ 2 C mL θ ⎠ → → 0 0 1 0 → → 0 d σ ( O) d σ ( O) 0 0 δ ( O) = = + Ω1 ∧ σ ( O) dt dt → → 0 δ ( O ) = R •• ⎛ ⎜ Aϕ 0 ⎜ ⎜ ⎜ ⎝ 1 ⎞ ⎟ ⎟ + ⎟ ⎛ 0⎞ ⎜ ⎟ ⎜ 0⎟∧ • ⎛ ⎜ Aϕ ⎜ 0 ⎜ ⎞ ⎟ ⎟ = ⎟ •• ⎛ Aϕ • • ( ) ( ) ( ) ⎟ ⎟⎟⎟ •• • • 2 ⎜ ⎟ 2 ⎟ ⎝ ⎠ ⎜ + ⎟ ⎜ •• C + mL θ θ θ 2 + θ ⎠ R C mL C mL 1 ⎝ ⎠ R ⎠ 1 ⎜ ⎜ ⎜ R ⎝ 1 Aϕθ ⎞ 4. L’énergie cinétique du système E C E C = = 1 2 1 2 ⎛ m⎜V ⎝ → 0 2 mL θ ⎞ ( O1 ) ⎟ ⎠ • 2 + 1 2 2 → 0T 2 1 + Ω 2 Aϕ • 2 + . I 1 Cθ 2 O1 • 2 → 0 2 . Ω = 1 2 • ⎛ ⎞ ⎡A 0 0⎤ ⎜ϕ ⎟ • • • 2 2 1⎛ ⎞ mL θ + ⎢ ⎥ ⎜ ⎟ ⎜ϕ, 0, θ ⎟ ⎝ ⎠ ⎢ 0 C 0 ⎥ 0 2 ⎜ • ⎟ R ⎢⎣ 0 0 C⎥⎦ ⎜θ ⎟ 1 R ⎝ ⎠ 1 5. Théorème de la résultante dynamique au système La somme des forces appliquées au système est égale à la masse du système par l’accélération de son centre d’inertie. → = → ∑ F 0 ext mγ ( O 1 ) ; avec : • ⎧ ⎪ − 2 Lθ → •• 0 γ ( O1 ) = ⎨ Lθ ⎪ 0 ⎪ R ⎩ 1 318
UMBB Boumerdès, Faculté des sciences, Département de physique Cours exercices, Mécanique Rationnelle : TCT et LMD-ST sem :3 A.KADI → R O → → + m g = mγ 0 ( O 1 ) ⇔ ⎧ ⎪ R ⎪ ⎨R ⎪R ⎪ ⎩ • 2 Ox + mg cosθ = −mLθ •• Oy Oz + R + R O1y O1z − mg sinθ = Lθ = 0 6. Théorème du moment dynamique du système au point O Le moment des forces extérieures est égal au moment dynamique au même point O. −→ → → M ext 0 ∑ 0 ( F ) = δ ( O) ⇔ OO 0 1∧ RO 1+ OO1 ∧ m g = δ ( O) −−→ → −−→ → → ⎛ L⎞ ⎛ 0 ⎜ ⎟ ⎜ ⎜ 0 ⎟∧ ⎜ RO ⎜ ⎟ ⎜ R ⎝ 0 ⎠ R ⎝ RO 1 1 1y 1z ⎞ ⎛ L⎞ ⎛ mg cosθ ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟+ ⎜ 0 ⎟∧ ⎜− mg sinθ ⎟= ⎟ ⎜ ⎟ ⎜ ⎟ ⎠ R ⎝ 0 ⎠ R ⎝ 0 ⎠ 1 1 •• ⎛ Aϕ • • ⎜ ⎜ ⎜ ⎜ R ⎝ 1 Aϕθ ⎞ •• 2 ( C + mL ) θ ⎟ ⎟⎟⎟⎟ ⎠ ⎛ 0 ⎜ ⎜− LR ⎜ ⎝ LR R O 1 • ϕ = Cte •• ⎛ ⎞ ⎜ Aϕ ⎟ ⎜ • • O 1z ⎟= ⎜ Aϕθ ⇔ − ⎟ •• θ ⎜ 2 ⎠ ( + ) θ ⎟ ⎟⎟⎟⎟ 1y mgLsin C mL R ⎝ ⎠ 1 ⎞ •• ⎧ ⎪ Aϕ = 0 • • ⎨− LRO 1z = Aϕθ ⎪ ⎪− mgLsinθ + LRO 1y ⎩ = 2 ( C + mL ) •• θ R O1 z A = − ϕθ L • • 2 ( C + mL )•• θ sinθ R O y + 1 = mg L Exercice : 04 Soit → → → R0 ( O, x0 , y0 , z0 ) un repère orthonormé fixe lié au bati d’une éolienne constitué d’une → → → 1( 1 1 1 girouette et d’une hélice. La girouette (S1) lié au repère R O, x , y , z ) , a une liaison pivot avec le bati fixe de manière à tourner dans le plan horizontal autour de l’axe → → → → −−→ → = 1 → → z 0 ≡ z 1 , α = ( x0 , x1 ) = ( y0 , y1) et OG a x où a : est une constante positive. → ( O, z0 ) , avec 319
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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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