Update textbook.fr.md

12.temporary_ins/40.classical-mechanics/30.n3/07.coherence/textbook.fr.md

@@ -523,6 +523,7 @@ $`\begin{align}
 &=\omega\;\overrightarrow{e_{\theta}}
 \end{align}`$
 
+-----------
 
 $`\begin{align}
 \dfrac{d\overrightarrow{e_{\theta}}}{dt}&=\dfrac{d}{dt}\big(-\sin\theta\;\overrightarrow{e_x}+\cos\theta\;\overrightarrow{e_z}\big)\\
@@ -535,21 +536,38 @@ $`\begin{align}
 \\
 &=\omega\;\Big(-\cos\theta\;\overrightarrow{e_x}-\sin\theta\;\overrightarrow{e_z}\Big)\\
 \\
-&=-\omega\;\overrightarrow{e_{\rho}}
+&=-\;\omega\;\overrightarrow{e_{\rho}}
 \end{align}`$
 
-
-
-
-
-
+--------------
 
 $`\begin{align}
-\overrrightarrow{\mathscr{v_M}}&=\dfrac{d\overrightarrow{OM}}{dt}=\dfrac{d}{dt}\left(\rho_M\overrightarrow{e_{\rho}\right)\\
-&=\dfrac{d\rho}{dt}\overrightarrow{e_(\rho)}+\rho\dfrac{d\\overrightarrow{e_{\rho}}{dt}\\
-&=\dfrac{d\rho}{dt}\overrightarrow{e_(\rho)}+\rho\dfrac{d\\overrightarrow{e_{\rho}}{dt}
+\dfrac{d\overrightarrow{e_{\theta}}}{dt}&=\dfrac{d}{dt}\big(-\sin\theta\;\overrightarrow{e_x}+\cos\theta\;\overrightarrow{e_z}\big)\\
+\\
+&=\Big[\dfrac{d(-\sin\theta)}{dt}\;\overrightarrow{e_x} - \sin\theta\;\underbrace{\dfrac{d\overrightarrow{e_z}}{dt}}_{=\vec{0}}\Big]\\
+&\quad\quad+\bigg[\dfrac{d\cos\theta}{dt}\;\overrightarrow{e_z}+\cos\theta\;\underbrace{\dfrac{d\overrightarrow{e_z}}{dt}}_{=\vec{0}}\bigg]\\
+\\
+&=\dfrac{d(-\sin\theta)}{d\theta}\;\underbrace{\dfrac{d\theta}{dt}}_{=\,\omega}\;\overrightarrow{e_x}
++\dfrac{d\cos\theta}{d\theta}\;\underbrace{\dfrac{d\theta}{dt}}_{=\,\omega}\;\overrightarrow{e_z}\\
+\\
+&=\omega\;\Big(-\cos\theta\;\overrightarrow{e_x}-\sin\theta\;\overrightarrow{e_z}\Big)\\
+\\
+&=-\;\omega\;\overrightarrow{e_{\rho}}
+\end{align}`$
 
+---------------
 
+$`\begin{align}
+\dfrac{d^2\overrightarrow{e_{\theta}}}{dt^2}&=\dfrac{d}{dt}\left(\underbrace{\dfrac{d\overrightarrow{e_{\rho}}}{dt}}_{=\omega\,\vec{e_{\theta}}\right)\\
+\\
+&=\dfrac{d}{dt}\left(\omega\,\overrightarrow{e_{\theta}}\right)\\
+\\
+&=\dfrac{d\omega}{dt}\;\overrightarrow{e_{\theta}}+\omega\;\dfrac{d\overrightarrow{e_{\theta}}}{dt}\\
+\\
+&=\dfrac{d\omega}{dt}\;\overrightarrow{e_{\theta}}+\omega\;\big(-\;\omega\;\overrightarrow{e_{\rho}}\big)\\
+\\
+&=\dfrac{d\omega}{dt}\;\overrightarrow{e_{\theta}}-\omega^2\;\overrightarrow{e_{\rho}}
+\end{align}