Update textbook.fr.md

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

@@ -510,15 +510,15 @@ $`\overrightarrow{e_{\theta}}=-\sin\theta\;\overrightarrow{e_x}+\cos\theta\;\ove
 
 
 $`\begin{align}
-\dfrac{d\overrightarrow{e_{\rho}}}{dt}&=\dfrac{d}{dt}\big(\cos\theta\;\overrightarrow{e_x}+\sin\theta\;\overrightarrow{e_z}\big)\\
+\dfrac{d\overrightarrow{e_{\rho}}}{dt}&=\dfrac{d}{dt}\bigg(\cos\theta\;\overrightarrow{e_x}+\sin\theta\;\overrightarrow{e_z}\bigg)\\
 \\
-&=\Big[\dfrac{d\cos\theta}{dt}\;\overrightarrow{e_x} + \sin\theta\;\underbrace{\dfrac{d\overrightarrow{e_z}}{dt}}_{=\vec{0}}\Big]\\
-&\quad\quad+\bigg[\dfrac{d(-\sin\theta)}{dt}\;\overrightarrow{e_z}+\cos\theta\;\underbrace{\dfrac{d\overrightarrow{e_z}}{dt}}_{=\vec{0}}\bigg]\\
+&=\bigg[\dfrac{d\cos\theta}{dt}\;\overrightarrow{e_x} + \sin\theta\;\underbrace{\dfrac{d\overrightarrow{e_z}}{dt}}_{=\;\vec{0}}\bigg]\\
+&\quad\quad+\bigg[\dfrac{d(-\sin\theta)}{dt}\;\overrightarrow{e_z}+\cos\theta\;\underbrace{\dfrac{d\overrightarrow{e_z}}{dt}}_{=\;\vec{0}}\bigg]\\
 \\
 &=\dfrac{d\cos\theta}{d\theta}\;\underbrace{\dfrac{d\theta}{dt}}_{=\,\omega}\;\overrightarrow{e_x}
 +\dfrac{d(-\sin\theta)}{d\theta}\;\underbrace{\dfrac{d\theta}{dt}}_{=\,\omega}\;\overrightarrow{e_z}\\
 \\
-&=\omega\;\Big(-\sin\theta\;\overrightarrow{e_x}+\cos\theta\;\overrightarrow{e_z}\Big)\\
+&=\omega\;\big(-\sin\theta\;\overrightarrow{e_x}+\cos\theta\;\overrightarrow{e_z}\big)\\
 \\
 &=\omega\;\overrightarrow{e_{\theta}}
 \end{align}`$
@@ -526,15 +526,15 @@ $`\begin{align}
 -----------
 
 $`\begin{align}
-\dfrac{d\overrightarrow{e_{\theta}}}{dt}&=\dfrac{d}{dt}\big(-\sin\theta\;\overrightarrow{e_x}+\cos\theta\;\overrightarrow{e_z}\big)\\
+\dfrac{d\overrightarrow{e_{\theta}}}{dt}&=\dfrac{d}{dt}\bigg(-\sin\theta\;\overrightarrow{e_x}+\cos\theta\;\overrightarrow{e_z}\bigg)\\
 \\
-&=\Big[\dfrac{d(-\sin\theta)}{dt}\;\overrightarrow{e_x} - \sin\theta\;\underbrace{\dfrac{d\overrightarrow{e_z}}{dt}}_{=\vec{0}}\Big]\\
-&\quad\quad+\bigg[\dfrac{d\cos\theta}{dt}\;\overrightarrow{e_z}+\cos\theta\;\underbrace{\dfrac{d\overrightarrow{e_z}}{dt}}_{=\vec{0}}\bigg]\\
+&=\bigg[\dfrac{d(-\sin\theta)}{dt}\;\overrightarrow{e_x} - \sin\theta\;\underbrace{\dfrac{d\overrightarrow{e_z}}{dt}}_{=\;\vec{0}}\bigg]\\
+&\quad\quad+\bigg[\dfrac{d\cos\theta}{dt}\;\overrightarrow{e_z}+\cos\theta\;\underbrace{\dfrac{d\overrightarrow{e_z}}{dt}}_{=\;\vec{0}}\bigg]\\
 \\
 &=\dfrac{d(-\sin\theta)}{d\theta}\;\underbrace{\dfrac{d\theta}{dt}}_{=\,\omega}\;\overrightarrow{e_x}
 +\dfrac{d\cos\theta}{d\theta}\;\underbrace{\dfrac{d\theta}{dt}}_{=\,\omega}\;\overrightarrow{e_z}\\
 \\
-&=\omega\;\Big(-\cos\theta\;\overrightarrow{e_x}-\sin\theta\;\overrightarrow{e_z}\Big)\\
+&=\omega\;\big(-\cos\theta\;\overrightarrow{e_x}-\sin\theta\;\overrightarrow{e_z}\big)\\
 \\
 &=-\;\omega\;\overrightarrow{e_{\rho}}
 \end{align}`$
@@ -558,7 +558,7 @@ $`\begin{align}
 ---------------
 
 $`\begin{align}
-\dfrac{d^2\overrightarrow{e_{\theta}}}{dt^2}&=\dfrac{d}{dt}\Big(\underbrace{\dfrac{d\overrightarrow{e_{\rho}}}{dt}}_{=\omega\,\vec{e_{\theta}}}\Big)\\
+\dfrac{d^2\overrightarrow{e_{\rho}}}{dt^2}&=\dfrac{d}{dt}\bigg(\underbrace{\dfrac{d\overrightarrow{e_{\rho}}}{dt}}_{=\omega\,\vec{e_{\theta}}}\bigg)\\
 \\
 &=\dfrac{d}{dt}\left(\omega\,\overrightarrow{e_{\theta}}\right)\\
 \\
@@ -569,6 +569,20 @@ $`\begin{align}
 &=\dfrac{d\omega}{dt}\;\overrightarrow{e_{\theta}}\;-\;\omega^2\;\overrightarrow{e_{\rho}}
 \end{align}`$
 
+---------------
+
+$`\begin{align}
+\dfrac{d^2\overrightarrow{e_{\theta}}}{dt^2}&=\dfrac{d}{dt}\bigg(\underbrace{\dfrac{d\overrightarrow{e_{\theta}}}{dt}}_{=-\omega\,\vec{e_{\rho}}}\bigg)\\
+\\
+&=\dfrac{d}{dt}\left(-\omega\,\overrightarrow{e_{\rho}}\right)\\
+\\
+&=-\dfrac{d\omega}{dt}\;\overrightarrow{e_{\rho}}\;-\;\omega\;\dfrac{d\overrightarrow{e_{\rho}}{dt}\\
+\\
+&=-\dfrac{d\omega}{dt}\;\overrightarrow{e_{\rho}}\;-\;\omega\;\big(\omega\;\overrightarrow{e_{\theta}}\big)\\
+\\
+&=\dfrac{d\omega}{dt}\;\overrightarrow{e_{\rho}}\;-\;\omega^2\;\overrightarrow{e_{\theta}}
+\end{align}`$
+
 
 
 _dynamique puis énergétique (énergie mécanique)_