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zFgqAj9FD?3#BCj6w&DmB5@HX*i$2&n*x{~}0B}B-69fuFJEFyrw)o+23=%AkK_MN* z9guJ&5{ZI=k&ccnO(lTmc>)*)$6o`0NKsHcnHL{grv@HUAq!LXf|%hsC3R z2Sy>GzZ(DVfhc_be_%-H9~VSn;J+u0LL>f65RLECf3No|J{XKbqmlmr|H3d}=7~eMSDW zkhA_{P-xg6S)<{oKZ2le2=Rg?-0ri6qlLtsi6H8d0sQAEhepj6pQE1^{&Ff>w4 SQ5B*JR)S-ca9a%k*Z%>X5=~D4 diff --git a/gravitation/potentiel_coulombien_distiller.tex b/gravitation/potentiel_coulombien_distiller.tex index 541a5b7..78c4fd1 100644 --- a/gravitation/potentiel_coulombien_distiller.tex +++ b/gravitation/potentiel_coulombien_distiller.tex @@ -154,7 +154,7 @@ r\sin\varphi\\ r^2 \dot{\varphi} \end{pmatrix} \] -Conservation du moment cin\'{e}tique demande que +Conservation du moment cin\'{e}tique demande que \[ \frac{\text{d}\vec{L}}{\text{d}t}=\vec{0} \] @@ -197,7 +197,7 @@ b(\vartheta)=\frac{k}{2E_0}\cot\left(\frac{\vartheta}{2}\right) \] \section{La distance minimale entre le particule $\alpha$ et le noyau d'or} -Le moment cin\'{e}tique en point $B$, qui a la distance minimale $r_B=r_\text{min}$ (l\`{a} $\vec{r}_B\perp \vec{v}_B$) est +Le moment cin\'{e}tique en point $B$, qui a la distance minimale $r_B=r_\text{min}$ (l\`{a} $\vec{r}_B\perp \vec{v}_B$) est \[ L_B=m_0r_Bv_B \] @@ -213,7 +213,7 @@ Avec $v_y(t_0)=0$, $v_y(t_B)=v_B\sin\left(\frac{\vartheta}{2}\right)$, $\varphi( \[ \left.\phantom{\frac{k}{bv_0}}m_0v_y\right|_0^{v_B\sin\left(\frac{\vartheta}{2}\right)}=\left.\frac{k}{bv_0}\cos\varphi\right|_\pi^{\frac{\pi+\vartheta}{2}} \] -on re\c{c}oit +on re\c{c}oit \[ m_0v_B\sin\left(\frac{\vartheta}{2}\right)=\frac{k}{bv_0}\left[\cos\left(\frac{\pi+\vartheta}{2}\right)+1\right] \] @@ -232,7 +232,7 @@ Soit $b=0$, le particule $\alpha$ mouve directement vers le noyau d'or sur l'axe \[ \frac{1}{2}m_0v_0^2=\frac{k}{r_0}+\frac{1}{2}m_0\underbrace{v_C^2}_{=0} \] -\c{C}a donne +\c{C}a donne \[ r_C=\frac{2k}{m_0v_0^2} \] @@ -240,8 +240,53 @@ et avec la notation de l'\'{e}nergie initiale $E_0=\frac{1}{2}m_0v_0^2$ on re\c{ \[ r_C=\frac{k}{E_0} \] + + +\section{L'enveloppe des trajectoires} + +\emph{La parabole est la courbe d'\'{e}quidistance entre un point (le foyer $F$) et une droite (la directrice $d$).} +\begin{center} +\begin{pspicture*}(-5,-4)(8,4) +\psaxes[ticks=none,labels=none,yAxis=false]{->}(0,0)(-4,-3.5)(7,3.5) +\psplot[linecolor=blue,plotpoints=500,polarplot]{-200}{200}{1 x cos sub 1 neg exp 2 mul} +\psdot(0,0) +\uput[135](0,0){$F$} +\psdot(-1,0) +\uput[135](-1,0){$A$} +\psline(-2,-3)(-2,3) +\uput[90](-2,3){Directrice $d$} +\psline[linestyle=dashed,linecolor=lightgray](0,0)(0,-3) +\psline[linestyle=dashed,linecolor=lightgray](-1,0)(-1,-2.5) +\psline{<->}(0,-2.25)(-1,-2.25) +\uput[90](-0.5,-2.25){$\frac{p}{2}$} +\psline{<->}(-2,-2.25)(-1,-2.25) +\uput[90](-1.5,-2.25){$\frac{p}{2}$} +\psline{<->}(0,-2.75)(-2,-2.75) +\uput[-90](-1,-2.75){$p$} +\psline[linecolor=red](0,0)(1,2.8)(-2,2.8) +\rput(0.8,1.3){\textcolor{red}{$x$}} +\uput[-90](-0.7,2.8){\textcolor{red}{$x$}} +\psdot[linecolor=red](1,2.8) +\uput[135](1,2.8){\textcolor{red}{$M$}} +%\psgrid +\end{pspicture*} +\end{center} +Une parabole en nommation polaire : +\[ +r(\varphi)=\frac{p}{1-\cos\varphi} +\] +Le param\`{e}tre $p$ est la distance du foyer $F$ de la parabole \`{a} la directrice $d$. + +L'enveloppe des trajectoires hyperboles est une parabole avec $p=2r_C$ : +\[ +r(\varphi)=\frac{2r_C}{1-\cos\varphi} +\] + + \newpage + \section{Les trajectoires des particules $\alpha$} + Les param\`{e}tres suivants de l'exp\'{e}rience originale sont : \begin{align*} m_0&=6,64\cdot 10^{-27}\,\text{kg}\\ @@ -259,28 +304,29 @@ v_{0x}&=2,1\cdot 10^7\,\text{m\,s}^{-1} \psaxes[ticks=none,labels=none]{->}(0,0)(-5,-8)(5,8) \uput[0](5,0){$x$} \uput[90](0,8){$y$} -\rput(3,1){Zone d'ombre} -\uput[135](-0.6,0){$C$} +\rput(3,0.3){Zone d'ombre} +%\uput[135](-0.6,0){$C$} \multido{\rA=-5+0.25}{41}{% \pstVerb{% /Pi 3.1415 def - /m0 6.64e-27 def +% /m0 6.64e-27 def + /m0 0.25 def /Z1 2 def /Z2 79 def /e0 1.6e-19 def /epsil 8.85e-12 def /COU1 Z1 Z2 mul e0 mul e0 mul 4 div Pi div epsil div m0 div def - /COU 7.5 def - /x0 6 neg def + /COU 5 m0 div def + /x0 200 neg def /y0 \rA\space def % /v0x 2.1e7 def - /v0x 5 def + /v0x 8 def /v0y 0 def /r1 COU 2 mul v0x 2 exp div neg def /facteur v0x dup mul COU div 4 div def }% \psequadiff[method=rk4, - plotpoints=1000, + plotpoints=2000, algebraic, whichabs=0, whichord=1, @@ -289,11 +335,8 @@ v_{0x}&=2,1\cdot 10^7\,\text{m\,s}^{-1} ]{0}{43}{x0 y0 v0x v0y}{\eqRuth}% \listplot[linecolor=red]{XiYi aload pop} } -\pscustom[fillstyle=solid,fillcolor=lightgray!30,linewidth=0pt,opacity=0.5]{% -\psplot[linecolor=lightgray,plotpoints=500]{r1}{6}{x r1 sub sqrt 1.45 mul} -\psline(!6 6 r1 sub sqrt 1.45 mul)(!6 6 r1 sub sqrt 1.45 mul neg) -\psplot[linecolor=lightgray,plotpoints=500]{6}{r1}{x r1 sub sqrt 1.45 mul neg} -} +\psplot[linecolor=blue,plotpoints=500,polarplot,fillstyle=solid,fillcolor=blue!40,opacity=0.25]{200}{-200}{1 x cos sub 1 neg exp facteur div} + \pscircle*[linecolor=yellow](0,0){0.3} \psdot(!r1 0) \end{pspicture*} -- 2.20.1