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 4.2. Les coordonnées cylindriques ( r, θ , z) Si le point P est repéré par les coordonnées cylindriques : ( r, θ , z) qui dépendent du temps → → → R r θ dans un repère ( O, u , u , k ) , le vecteur position s’écrirait : −→ → → OP = r u + z k r −→ → d OP • → d u • → r V = = r ur + r + z k dt dt avec → d u dt → • → • → V r ur + rθ uθ V r → → d u • → r dθ = . = θ uθ , on obtient ainsi : dθ dt • → = + z k • • • r = r , Vθ = r θ , Vz = z → x → z o r θ P(t) z → u θ → u r → y → → → Dans le repère R( O, i , j, k) le vecteur OP s’écrit : L’accélération est déterminée par : −→ ⎧r cosθ −→ ⎪ OP = ⎨r sinθ ⎪ ⎩z −→ → • → • → → 2 d OP d V d( r u d r u •• → r ) ( θ θ ) γ = = = + + z k 2 dt dt dt dt → → → •• → • d u • • → •• → • •• → r d uθ γ = r ur + r + rθ uθ + rθ uθ + rθ + z k dt dt or nous avons : → d u dt r → d u • → r dθ = . = θ uθ ; dθ dt L’expression de l’accélération devient : → d u dt θ → d u • → θ dθ = . = −θ ur dθ dt → •• • → •• • • → 2 r− rθ ) ur + ( rθ + 2rθ uθ •• → γ = ( ) + z k d’où : •• • 2 2 • • 2 •• 2 γ = ( r− rθ ) + ( rθ + 2rθ ) + z •• •• • •• • • •• 2 γ r = ( r− rθ ) ; γ θ = ( rθ + 2rθ ) ; γ z = z 4.3. Les coordonnées sphériques ( r , θ , ϕ) ⎧r cosϕ cosθ → → → −→ −→ ⎪ Dans le repère R( O, i , j, k) le vecteur OP a pour composantes : OP = ⎨r cosϕ sinθ ⎪ ⎩r sinϕ En coordonnées sphériques il s’écrit : OP −→ → → = OP.er = r er 184
UMBB Boumerdès, Faculté des sciences, Département de physique Cours exercices, Mécanique Rationnelle : TCT et LMD-ST sem :3 A.KADI → e ϕ → z ϕ ϕ → e r → u → z o → e ϕ r ϕ → e r → e θ → y avec : → x θ → e θ → u → e r → → = cos ϕ u+ sinϕ k ; e → ϕ → → = −sinϕ u+ cosϕ k → → → u = ϕ er − sinϕ eϕ cos ; → → = e θ d u dθ ; → d e → θ = − u dθ ; → e → = e ϕ → d r d e → ϕ ; = −er dϕ dϕ alors : → d e r dt → • → d u • → • → • → • → = − ϕ sin ϕ u+ cos ϕ + ϕ cos ϕ k = θ cos ϕ eθ − ϕ sin ϕ u+ ϕ cos ϕ k dt → d e r dt • → • → → • → • → = θ ϕ eθ + ϕ ( − sin ϕ u+ cos ϕ k ) = θ cos ϕ eθ + ϕ eϕ cos ; on déduit la vitesse du point P : −→ → → → d OP d( r e • → • → • → • → r ) d er V = = = r er + r = r er + rθ cosϕ eθ + rϕ eϕ dt dt dt → ; ⇒ V ⎧ ⎪ = ⎨V ⎪ ⎪V ⎩ • V r = r • θ = rθ cosϕ • ϕ = rϕ l’accélération se déduit facilement en dérivant l’expression de la vitesse par rapport au temps : → γ = −→ • → • → • → d V d( r e d r e d( r e ) r ) ( θ cosϕ ) ϕ θ ϕ = + + dt dt dt dt (1) : • → d( r er ) dt •• → • • → • → •• → • • → • • → = r er + r( θ cosϕ eθ + ϕ eϕ ) = r er + rθ cosϕ eθ + rϕ eϕ (2) : • → d( rθ cosϕ eθ dt ) • • → •• → • • → • d eθ = rθ cosϕ eθ + rθ cosϕ eθ − rθ ϕ sinϕ eθ + rθ cosϕ dt → d e dt → θ → d e • → • → → θ dθ = . = −θ u = −θ (cosϕ er − sinϕ eϕ ) dθ dt • → • d( rθ cosϕ e ) • • → •• → • • → → → θ 2 = rθ cosϕ eθ + rθ cosϕ eθ − rθ ϕ sinϕ eθ − rθ cosϕ(cosϕ er − sinϕ eϕ ) dt 185
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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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