Update textbook.fr.md

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

@@ -517,14 +517,21 @@ $`\begin{align}
 &=\bigg[\dfrac{d\cos\theta}{dt}\;\overrightarrow{e_x} + \sin\theta\;\underbrace{\dfrac{d\overrightarrow{e_z}}{dt}}_{=\,\vec{0}}\bigg]\\
 &\quad\quad+\bigg[\dfrac{d(-\sin\theta)}{dt}\;\overrightarrow{e_z}+\cos\theta\;\underbrace{\dfrac{d\overrightarrow{e_z}}{dt}}_{=\,\vec{0}}\bigg]\\
 \\
-&=\dfrac{d\cos\theta}{d\theta}\;\underbrace{\dfrac{d\theta}{dt}}_{=\,\omega}\;\overrightarrow{e_x}
-+\dfrac{d(-\sin\theta)}{d\theta}\;\underbrace{\dfrac{d\theta}{dt}}_{=\,\omega}\;\overrightarrow{e_z}\\
+&=\dfrac{d\cos\theta}{d\theta}\;\dfrac{d\theta}{dt}\;\overrightarrow{e_x}
++\dfrac{d(-\sin\theta)}{d\theta}\;\dfrac{d\theta}{dt}\;\overrightarrow{e_z}\\
 \\
-&=\omega\;\big(-\sin\theta\;\overrightarrow{e_x}+\cos\theta\;\overrightarrow{e_z}\big)\\
+&=\dfrac{d\theta}{dt}\;\big(-\sin\theta\;\overrightarrow{e_x}+\cos\theta\;\overrightarrow{e_z}\big)\\
 \\
-&=\omega\;\overrightarrow{e_{\theta}}
+&=\dfrac{d\theta}{dt}\;\overrightarrow{e_{\theta}}
 \end{align}`$
 
+Note : Dans le cas général, on évite d'écrire $`\dfrac{d\theta}{dt}=\omega`$ où $`\omega`$ 
+est la pulsation d'unité SI $`rad\,s-{-1}`$.   
+L'écriture $`\dfrac{d\theta}{dt}=\omega`$ est réservée au cas où la variation temporelle de 
+l'angle $`\theta`$ a une composante sinusoïdale de pulsation $`\omega`$ stationnaire.
+
+\dfrac{d\theta}{dt}
+
 --------------
 
 $`\begin{align}
@@ -533,41 +540,41 @@ $`\begin{align}
 &=\bigg[\dfrac{d(-\sin\theta)}{dt}\;\overrightarrow{e_x} - \sin\theta\;\underbrace{\dfrac{d\overrightarrow{e_z}}{dt}}_{=\,\vec{0}}\bigg]\\
 &\quad\quad+\bigg[\dfrac{d\cos\theta}{dt}\;\overrightarrow{e_z}+\cos\theta\;\underbrace{\dfrac{d\overrightarrow{e_z}}{dt}}_{=\,\vec{0}}\bigg]\\
 \\
-&=\dfrac{d(-\sin\theta)}{d\theta}\;\underbrace{\dfrac{d\theta}{dt}}_{=\,\omega}\;\overrightarrow{e_x}
-+\dfrac{d\cos\theta}{d\theta}\;\underbrace{\dfrac{d\theta}{dt}}_{=\,\omega}\;\overrightarrow{e_z}\\
+&=\dfrac{d(-\sin\theta)}{d\theta}\;\dfrac{d\theta}{dt}\;\overrightarrow{e_x}
++\dfrac{d\cos\theta}{d\theta}\;\dfrac{d\theta}{dt}\;\overrightarrow{e_z}\\
 \\
-&=\omega\;\Big(-\cos\theta\;\overrightarrow{e_x}-\sin\theta\;\overrightarrow{e_z}\Big)\\
+&=\dfrac{d\theta}{dt}\;\Big(-\cos\theta\;\overrightarrow{e_x}-\sin\theta\;\overrightarrow{e_z}\Big)\\
 \\
-&=-\;\omega\;\overrightarrow{e_{\rho}}
+&=-\;\dfrac{d\theta}{dt}\;\overrightarrow{e_{\rho}}
 \end{align}`$
 
 ---------------
 
 $`\begin{align}
-\dfrac{d^2\overrightarrow{e_{\rho}}}{dt^2}&=\dfrac{d}{dt}\bigg(\underbrace{\dfrac{d\overrightarrow{e_{\rho}}}{dt}}_{=\,\omega\,\vec{e_{\theta}}}\bigg)\\
+\dfrac{d^2\overrightarrow{e_{\rho}}}{dt^2}&=\dfrac{d}{dt}\bigg(\underbrace{\dfrac{d\overrightarrow{e_{\rho}}}{dt}}_{=\,\dfrac{d\theta}{dt}\,\vec{e_{\theta}}}\bigg)\\
 \\
-&=\dfrac{d}{dt}\left(\omega\,\overrightarrow{e_{\theta}}\right)\\
+&=\dfrac{d}{dt}\left(\dfrac{d\theta}{dt}\,\overrightarrow{e_{\theta}}\right)\\
 \\
-&=\dfrac{d\omega}{dt}\;\overrightarrow{e_{\theta}}\;+\;\omega\;\dfrac{d\overrightarrow{e_{\theta}}}{dt}\\
+&=\dfrac{d^2\theta}{dt^2}\;\overrightarrow{e_{\theta}}\;+\;\dfrac{d\theta}{dt}\;\dfrac{d\overrightarrow{e_{\theta}}}{dt}\\
 \\
-&=\dfrac{d\omega}{dt}\;\overrightarrow{e_{\theta}}\;+\;\omega\;\big(-\,\omega\;\overrightarrow{e_{\rho}}\big)\\
+&=\dfrac{d^2\theta}{dt^2}\;\overrightarrow{e_{\theta}}\;+\;\dfrac{d\theta}{dt}\;\left(-\,\dfrac{d\theta}{dt}\;\overrightarrow{e_{\rho}}\right)\\
 \\
-&=\dfrac{d\omega}{dt}\;\overrightarrow{e_{\theta}}\;-\;\omega^2\;\overrightarrow{e_{\rho}}
+&=\dfrac{d^2\theta}{dt^2}\;\overrightarrow{e_{\theta}}\;-\;\left(\dfrac{d\theta}{dt}\right)^2\;\overrightarrow{e_{\rho}}
 \end{align}`$
 
 ---------------
 
 $`\begin{align}
 \dfrac{d^2\overrightarrow{e_{\theta}}}{dt^2}
-&=\dfrac{d}{dt}\bigg(\underbrace{\dfrac{d\overrightarrow{e_{\theta}}}{dt}}_{=-\omega\,\vec{e_{\rho}}}\bigg)\\
+&=\dfrac{d}{dt}\bigg(\underbrace{\dfrac{d\overrightarrow{e_{\theta}}}{dt}}_{=-\dfrac{d\theta}{dt}\,\vec{e_{\rho}}}\bigg)\\
 \\
-&=\dfrac{d}{dt}\left(-\omega\,\overrightarrow{e_{\rho}}\right)\\
+&=\dfrac{d}{dt}\left(-\dfrac{d\theta}{dt}\,\overrightarrow{e_{\rho}}\right)\\
 \\
-&=-\dfrac{d\omega}{dt}\;\overrightarrow{e_{\rho}}\;-\;\omega\;\dfrac{d\overrightarrow{e_{\rho}}}{dt}\\
+&=-\dfrac{d^2\theta}{dt^2}\;\overrightarrow{e_{\rho}}\;-\;\dfrac{d\theta}{dt}\;\dfrac{d\overrightarrow{e_{\rho}}}{dt}\\
 \\
-&=-\dfrac{d\omega}{dt}\;\overrightarrow{e_{\rho}}\;-\;\omega\;\big(\omega\;\overrightarrow{e_{\theta}}\big)\\
+&=-\dfrac{d^2\theta}{dt^2}\;\overrightarrow{e_{\rho}}\;-\;\dfrac{d\theta}{dt}\;\big(\omega\;\overrightarrow{e_{\theta}}\big)\\
 \\
-&=\dfrac{d\omega}{dt}\;\overrightarrow{e_{\rho}}\;-\;\omega^2\;\overrightarrow{e_{\theta}}
+&=\dfrac{d^2\theta}{dt^2}\;\overrightarrow{e_{\rho}}\;-\;\left(\dfrac{d\theta}{dt}\right)^2\;\overrightarrow{e_{\theta}}
 \end{align}`$
 
 ----------------
@@ -583,20 +590,20 @@ $`\begin{align}
 \\
 &=\underbrace{\dfrac{d\mathscr{l}}{dt}}_{=\,0}\;\overrightarrow{e_{\rho}}\;+\;\mathscr{l}\;\dfrac{d\overrightarrow{e_{\rho}}}{dt}\\
 \\
-&=\mathscr{l}\;\omega\;\overrightarrow{e_{\theta}}
+&=\mathscr{l}\;\dfrac{d\theta}{dt}\;\overrightarrow{e_{\theta}}
 \end{align}`$
 
 
 $`\begin{align}
 \overrightarrow{a_M}&=\dfrac{d\overrightarrow{\mathscr{v}_M}}{dt}\\
 \\
-&=\dfrac{d}{dt}\big(\mathscr{l}\;\omega\;\overrightarrow{e_{\theta}}\big)\\
+&=\dfrac{d}{dt}\left(\mathscr{l}\;\dfrac{d\theta}{dt}\;\overrightarrow{e_{\theta}}\right)\\
 \\
-&=\underbrace{\dfrac{d\mathscr{l}}{dt}}_{=\,0}\;\omega\;\overrightarrow{e_{\theta}}\;
-+\;\mathscr{l}\;\dfrac{d\omega}{dt}\;\overrightarrow{e_{\theta}}\;
-+\;\mathscr{l}\;\omega\;\underbrace{\dfrac{d\overrightarrow{e_{\theta}}}{dt}}_{=\,-\omega\,\vec{e_{\rho}}}\\
+&=\underbrace{\dfrac{d\mathscr{l}}{dt}}_{=\,0}\;\dfrac{d\theta}{dt}\;\overrightarrow{e_{\theta}}\;
++\;\mathscr{l}\;\dfrac{d^2\theta}{dt^2}\;\overrightarrow{e_{\theta}}\;
++\;\mathscr{l}\;\dfrac{d\theta}{dt}\;\underbrace{\dfrac{d\overrightarrow{e_{\theta}}}{dt}}_{=\,-\dfrac{d\theta}{dt}\,\vec{e_{\rho}}}\\
 \\
-&=\mathscr{l}\;\dfrac{d\omega}{dt}\;\overrightarrow{e_{\theta}}\;-\;\mathscr{l}\;\omega^2\overrightarrow{e_{\rho}}
+&=\mathscr{l}\;\dfrac{d^2\theta}{dt^2}\;\overrightarrow{e_{\theta}}\;-\;\mathscr{l}\;\left(\dfrac{d\theta}{dt}\right)^2\overrightarrow{e_{\rho}}
 \end{align}`$
 
 
@@ -634,13 +641,13 @@ La masse du corps M est constante.
 
 $`m\;\overrightarrow{a_M}\cdot\overrightarrow{e_{\rho}}=\overrightarrow{F_{totale}}\cdot\overrightarrow{e_{\rho}}`$
 
-$`\Longrightarrow\quad -\;\mathscr{l}\;\omega^2=m\,g\,\cos\theta-R`$
+$`\Longrightarrow\quad -\;\mathscr{l}\;\left(\dfrac{d\theta}{dt}\right)^2=m\,g\,\cos\theta-R`$
 
 Projetons la deuxième loi de Newtion sur $`\overrightarrow{e_{\theta}}`$ :
 
 $`m\;\overrightarrow{a_M}\cdot\overrightarrow{e_{\theta}}=\overrightarrow{F_{totale}}\cdot\overrightarrow{e_{\theta}}`$
 
-$`\Longrightarrow\quad\mathscr{l}\;\dfrac{d\omega}{dt}=-\,m\,g\,\sin\theta`$
+$`\Longrightarrow\quad\mathscr{l}\;\dfrac{d^2\theta}{dt^2}=-\,m\,g\,\sin\theta`$