183 lines
10 KiB
TeX
183 lines
10 KiB
TeX
\documentclass[a4paper,10pt]{article}
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%\documentclass[a4paper,10pt]{scrartcl}
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\input{common.tex}
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\begin{document}
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\begin{tabularx}{\textwidth}{ |X|X|X| }
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\hline
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\textbf{Produits vectoriels} \newline
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$ \vec{e}_x \times \vec{e}_y = -\vec{e}_y \times \vec{e}_x = \vec{e}_z $ \newline
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$ \vec{e}_y \times \vec{e}_z = -\vec{e}_z \times \vec{e}_y = \vec{e}_x $ \newline
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$ \vec{e}_z \times \vec{e}_x = -\vec{e}_x \times \vec{e}_z = \vec{e}_y $ \newline
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$ \vec{e}_x \times \vec{e}_x = \vec{e}_y \times \vec{e}_y = \vec{e}_z \times \vec{e}_z = \vec{0} $ \newline
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&
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\textbf{MRUA} \newline
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$ r = \frac{1}{2} \cdot a_0 \cdot t^2 + v_0 \cdot t + r_0 $ \newline
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$ v = a_0 \cdot t + v_0 $ \newline
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$ a = a_0 $ \newline
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&
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\textbf{MCU} \newline
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$ a = \frac{v^2}{r} = \omega^2 \cdot r$ \newline
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$ \vec{v} = \vec{\omega} \times \vec{r} $ \newline
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$ \vec{a} = \vec{\alpha} \times \vec{r} $ \newline
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$ \omega \cdot T = 2 \cdot \pi $ \newline
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\\ \hline
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\textbf{Moments / Centre de masse} \newline
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$ \vec{L}_O = \vec{r} \times \vec{p} = m \cdot \vec{r} \times \vec{v} $ \newline
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$ \vec{M}_O = \vec{r} \times \vec{F} = \frac{\mathrm{d}\vec{L}_O}{\mathrm{d}t} $ \newline
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$ \vec{r}_{cm} = \frac{1}{M} \int_{M} \vec{r} \cdot \mathrm{d}m = \frac{1}{M} \int_{V} \vec{r} \cdot \rho(\vec{r}) \cdot \mathrm{d}V $ \newline
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$ I_{cm,\Delta} = \int_{M} r_\bot^2 \cdot \mathrm{d}m $ \newline
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$ \vec{L}_{cm,\Delta} = I_{cm,\Delta} \cdot \vec{\omega} $ \newline
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$ \vec{M}_{cm,\Delta} = I_{cm,\Delta} \cdot \vec{\alpha} $ \newline
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$ I = I_{cm} + M \cdot r^2 $ \newline
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$ \vec{r}_{cm} = \frac{1}{M} \sum m_i \cdot \vec{r}_i $ \newline
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&
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\textbf{Forces} \newline
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$ \vec{p} = m \cdot \vec{v} $ \newline
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$ \vec{F} = m \cdot \vec{a} = \frac{\mathrm{d}\vec{p}}{\mathrm{d}t} $ \newline
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$ \vec{F}_f = \mu \cdot \vec{N} $ \newline
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$ \vec{F}_f = -K \cdot \eta \cdot \vec{v} $ \newline
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$ W = \int \vec{F} \bullet \mathrm{d}\vec{r} $ \newline
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$ P_{inst} = \frac{\mathrm{d}W}{\mathrm{d}t} = \vec{F} \bullet \vec{v} $ \newline
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$ P_{moy} = \frac{W}{\Delta t} $ \newline
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&
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\textbf{Énergie} \newline
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$ W = \Delta E $ \newline
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$ E_{mec} = E_{cin} + E_{pot} $ \newline
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$ E_{mec,sat} = - \frac{G \cdot M \cdot m}{2 \cdot r} $ \newline
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$ E_{cin} = \frac{1}{2} \cdot m \cdot v^2 $ \newline
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$ E_{cin} = \frac{1}{2} \cdot m \cdot \omega_0^2 \cdot (A^2 - x^2) $ \newline
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$ E_{cin} = \frac{1}{2} \cdot I_{cm,\Delta} \cdot \omega^2 $ \newline
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$ E_{pot} = m \cdot g \cdot h $ \newline
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$ E_{pot} = \frac{1}{2} \cdot k \cdot x^2 = \frac{1}{2} \cdot m \cdot \omega_0^2 \cdot x^2 $ \newline
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$ E_{pot} = - \frac{G \cdot M \cdot m}{r} $ \newline
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\\ \hline
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\textbf{Référentiel non-galiléen} \newline
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$ m \cdot \vec{a}' = \sum \vec{F}_{ext} - m \cdot \vec{a}_e - m \cdot \vec{a}_{Cor} $ \newline
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$ - m \cdot \vec{a}_e = - m \cdot \vec{\omega} \times (\vec{\omega} \times \vec{r})$ \newline
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$ - m \cdot \vec{a}_{Cor} = - 2 \cdot m \cdot \vec{\omega} \times \vec{v}' $ \newline
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\textbf{Balistique} \newline
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$ h_{max} = \frac{(v_0 \cdot \sin(\alpha))^2}{2 \cdot g} $ \newline
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$ p = \frac{v_0^2 \cdot \sin(2 \cdot \alpha)}{g} $ \newline
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\textbf{Intégrales volumiques} \newline
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$ V = \iiint\limits_{cube} \mathrm{d}V = \iiint \mathrm{d}x \cdot \mathrm{d}y \cdot \mathrm{d}z $ \newline
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$ V = \iiint\limits_{cylindre} \mathrm{d}V = \iiint \rho \cdot \mathrm{d}\rho \cdot \mathrm{d}\varphi \cdot \mathrm{d}z $ \newline
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$ V = \iiint\limits_{boule} \mathrm{d}V = \iiint r^2 \cdot \sin(\theta) \cdot \mathrm{d}r \cdot \mathrm{d}\theta \cdot \mathrm{d}\varphi $ \newline
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\\ \hline
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\textbf{Kepler} \newline
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$ \frac{a^3}{T^2} = \frac{G \cdot M}{4 \cdot \pi^2} $ \hfill 1\textsuperscript{ère} loi \newline
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$ \frac{\mathrm{d}\vec{A}}{\mathrm{d}t} = \frac{1}{2} \cdot \vec{r} \times \vec{v} = \frac{\vec{L}_O}{2 \cdot m} $ \hfill 2\textsuperscript{ème} loi \newline
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$ \vec{F} = - \frac{G \cdot M \cdot m}{r^2} \cdot \vec{u_r} $ \hfill 3\textsuperscript{ème} loi \newline
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$ T = 2 \cdot \pi \sqrt{\frac{R^3}{G \cdot M}} $ \newline
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&
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\textbf{Dérivées usuelles} \newline
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$ v = \dot{r} $ \newline
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$ a = \dot{v} = \ddot{r} $ \newline
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$ \omega = \dot{\varphi} $ \newline
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$ \alpha = \dot{\omega} = \ddot{\varphi} $ \newline
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$ F = \dot{p} $ \newline
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$ P = \dot{W} $ \newline
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$ M = \dot{L} $ \newline
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&
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\textbf{} \newline
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\includegraphics[width=0.25\textwidth,keepaspectratio=true]{./sys_coord.png} \newline
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\\ \hline
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\textbf{Ressort / Pendule} \newline
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$ \vec{F} = -k \cdot \vec{r} = -k \cdot (\vec{l} - \vec{l}_0) $ \hfill (ressort) \newline
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$ T_0 = \frac{2 \cdot \pi}{\omega_0} $ \newline
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$ f_0 = \frac{1}{T_0} = \frac{\omega_0}{2 \cdot \pi} $ \newline
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$ \omega_0 = \sqrt{\frac{k}{m}} \text{ ou } \omega_0 = \sqrt{\frac{g}{l}} $ \newline
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$ \ddot{x} + \omega_0^2 \cdot x = 0 $ \newline
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$ x(t) = A_1 \cdot \cos(\omega_0 \cdot t + \Phi) $ \newline
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&
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\textbf{Oscillateurs} \newline
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$ \ddot{x} + 2 \cdot \lambda \cdot \dot{x} + \omega_0^2 \cdot x = 0 \mid x = C \cdot e^{\gamma \cdot t} $ \newline
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$ \gamma = - \lambda \pm \sqrt{\lambda^2 - \omega_0^2} $ \newline
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$ \omega = \sqrt{| \omega_0^2 - \lambda^2 |} $ \newline
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$ x(t) = A \cdot e^{- \lambda \cdot t} \cdot \cos(\omega \cdot t + \Phi), $ \hfill $ \lambda^2 < \omega_0^2 $ \newline
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$ x(t) = e^{- \lambda \cdot t} \cdot (A_1 \cdot e^{\omega \cdot t} + A_2 \cdot e^{-\omega \cdot t}), $ \hfill $ \lambda^2 > \omega_0^2 $ \newline
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$ x(t) = (A + B \cdot t) \cdot e^{- \lambda \cdot t}, $ \hfill $ \lambda^2 = \omega_0^2 $ \newline
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\textbf{Oscillateurs forcés} \newline
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$ \ddot{x} + 2 \cdot \lambda \cdot \dot{x} + \omega_0^2 \cdot x = f \cdot \cos(\Omega \cdot t) $ \newline
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$ x = A(\Omega) \cdot \cos(\Omega \cdot t + \psi) $ \newline
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$ \underline{x} = A(\Omega) \cdot e^{i \cdot \psi(\Omega)} \cdot e^{i \cdot \Omega \cdot t} = x_0 \cdot e^{i \cdot \Omega \cdot t} $ \newline
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$ \omega_0 = \sqrt{\frac{k}{m}}, \lambda = \frac{\chi}{2 \cdot m}, f = \frac{F_e}{m} $ \newline
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$ \omega = \sqrt{w_0^2 - \lambda^2}$ \newline
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$ x_0 = A(\Omega) \cdot e^{i \cdot \psi(\Omega)} = \frac{f}{\omega_0^2 - \Omega^2 + i \cdot 2 \cdot \lambda \cdot \Omega} $ \newline
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$ A(\Omega) = \|x_0\| = \frac{f}{\sqrt{(\omega_0^2 - \Omega^2)^2 + (2 \cdot \lambda \cdot \Omega)^2}} $ \newline
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$ \psi(\Omega) = \arctan(\frac{\Im(x_0)}{\Re(x_0)}) = \arctan(\frac{-2 \cdot \lambda \cdot \Omega}{\omega_0^2 - \Omega^2}) $ \newline
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$ \Omega_r = \sqrt{w_0^2 - 2 \cdot \lambda^2} $ \hfill $ \frac{\mathrm{d}A(\Omega)}{\mathrm{d}\Omega} = 0 $ \newline
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$ Q = \frac{\Omega_r}{\Delta \Omega} = \frac{\Omega_r^2}{2 \cdot \lambda \cdot \omega} $ \newline
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\\ \hline
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\textbf{Coordonnées polaires $ (O,\vec{e_r},\vec{e}_{\varphi}) $} \newline
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$ \vec{r} = r \cdotbis \vec{e_r} $ \newline
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$ \vec{v} = \dot{r} \cdotbis \vec{e_r} + r \cdotbis \dot{\varphi} \cdotbis \vec{e}_{\varphi} $ \newline
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$ \vec{a} = (\ddot{r} - r \cdotbis \dot{\varphi}^2) \cdotbis \vec{e_r} + (r \cdotbis \ddot{\varphi} + 2 \cdotbis \dot{r} \cdotbis \dot{\varphi}) \cdotbis \vec{e}_{\varphi} $ \newline
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$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e_r} = \dot{\varphi} \cdotbis \vec{e}_{\varphi} $ \newline
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$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e}_{\varphi} = -\dot{\varphi} \cdotbis \vec{e_r} $ \newline
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\textbf{Coord. cylindriques $ (O,\vec{e}_{\rho},\vec{e}_{\varphi},\vec{e}_z) $} \newline
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$ \vec{r} = \rho \cdotbis \vec{e}_{\rho} + z \cdotbis \vec{e}_z $ \newline
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$ \vec{v} = \dot{\rho} \cdotbis \vec{e}_{\rho} + \rho \cdotbis \dot{\varphi} \cdotbis \vec{e}_{\varphi} + \dot{z} \cdotbis \vec{e}_z $ \newline
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$ \vec{a} = (\ddot{\rho} - \rho \cdotbis \dot{\varphi}^2) \cdotbis \vec{e}_{\rho} + (\rho \cdotbis \ddot{\varphi} + 2 \cdotbis \dot{\rho} \cdotbis \dot{\varphi}) \cdotbis \vec{e}_{\varphi} + \ddot{z} \cdotbis \vec{e}_z $ \newline
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$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e}_{\rho} = \dot{\varphi} \cdotbis \vec{e}_{\varphi} $ \newline
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$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e}_{\varphi} = -\dot{\varphi} \cdotbis \vec{e}_{\rho} $ \newline
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$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e}_z = 0 $ \newline
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\textbf{Coord. sphériques $ (O,\vec{e}_{r},\vec{e}_{\theta},\vec{e}_{\varphi}) $} \newline
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$ \vec{r} = r \cdotbis \vec{e_r}$ \newline
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$ \vec{v} = \dot{r} \cdotbis \vec{e_r} + r \cdotbis \dot{\theta} \cdotbis \vec{e}_{\theta} + r \cdotbis \dot{\varphi} \cdotbis \sin(\theta) \cdotbis \vec{e}_{\varphi} $ \newline
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$ \vec{a} = \begin{pmatrix}
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\ddot{r} - \dot{r} \cdotbis \dot{\theta}^2 - r \cdotbis \dot{\varphi}^2 \cdotbis \sin^2(\theta) \\
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2 \cdotbis \dot{r} \cdotbis \dot{\theta} + r \cdotbis \ddot{\theta} - r \cdotbis \dot{\varphi}^2 \cdotbis \sin(\theta) \cdotbis \cos(\theta) \\
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2 \cdotbis \dot{r} \cdotbis \dot{\varphi} \cdotbis \sin(\theta) + r \cdotbis \ddot{\varphi} \cdotbis \sin(\theta) + 2 \cdotbis r \cdotbis \dot{\varphi} \cdotbis \dot{\theta} \cdotbis \cos(\theta) \\
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\end{pmatrix} $ \newline
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$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e_r} = \dot{\theta} \cdotbis \vec{e}_{\theta} + \dot{\varphi} \cdotbis \sin(\theta) \cdotbis \vec{e}_{\varphi} $ \newline
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$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e}_{\theta} = -\dot{\theta} \cdotbis \vec{e_r} + \dot{\varphi} \cdotbis \cos(\theta) \cdotbis \vec{e}_{\varphi} $ \newline
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$ \frac{\mathrm{d}}{\mathrm{d}t} \cdotbis \vec{e}_{\varphi} = -\dot{\varphi} \cdotbis \sin(\theta) \cdotbis \vec{e_r} - \dot{\varphi} \cdotbis \cos(\theta) \cdotbis \vec{e}_{\theta} $ \newline
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\\ \hline
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\textbf{Équations de base} \newline
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$ \sum \vec{F} = m \cdot \vec{a} $ \newline
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$ \sum \vec{M}_O = \frac{\mathrm{d}}{\mathrm{d}t} \vec{L}_O $ \newline
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$ \sum \vec{p} = cte $ \newline
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$ E_i - E_f = 0 $ \newline
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&
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\textbf{Signes} \newline
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$ r, v, a, \omega, \alpha, F $ \hfill avec \newline
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$ M, L, p $ \hfill sans \newline
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&
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\textbf{Angles} \newline
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$ \cos(\pi \pm \alpha) = - \cos(\alpha) $ \newline
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$ \cos(\frac{\pi}{2} + \alpha) = - \sin(\alpha) $ \newline
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$ \cos(\frac{\pi}{2} - \alpha) = \sin(\alpha) $ \newline
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$ \sin(\pi + \alpha) = - \sin(\alpha) $ \newline
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$ \sin(\pi - \alpha) = \sin(\alpha) $ \newline
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$ \sin(\frac{\pi}{2} \pm \alpha) = \cos(\alpha) $ \newline
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\\ \hline
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% &
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% \textbf{Configurabilité} \newline
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% $ a \oldcdot b $ ou $ a b$ \newline
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% $ \frac{a}{b} $ ou $ a/b$ \newline
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% $ \vec{a} \oldbullet \vec{b} $ ou $ \vec{a} \circ \vec{b} $ \newline
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% $ \oldvec{a} $ ou $ \overrightarrow{a} $ ou $ \mathbf{a} $ ou $ \oldvec{\mathbf{a}} $ \newline
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% $ \dot{x} $ ou $ \frac{\mathrm{d}x}{\mathrm{d}t} $ \newline
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% $ \ddot{x} $ ou $ \frac{\mathrm{d^2}x}{\mathrm{d}t^2} $ \newline
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% &
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% \\ \hline
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\end{tabularx}
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\end{document}
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