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\begin{document}
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\title{Mécanique du Solide}
\author{Aubin SIONVILLE}
\date{MPI Clemenceau - 2021-2023}
\maketitle

\centering
\section*{Rotations}
\begin{multicols}{2}
    \subsection*{Vecteur instantané de rotation}
    \[\overrightarrow{\Omega} = \dot{\theta}\overrightarrow{u_\Delta}\]

    \subsection*{Vitesse de rotation}
    \[\overrightarrow{v} = \overrightarrow{\Omega} \wedge \overrightarrow{OM}\]
\end{multicols}

\begin{multicols}{2}
    
    \begin{tikzpicture}
        % Axe vertical
        \draw[->] (0,-2) -- (0,2);
        % Bras de de levier perpendiculaire à l'axe
        \draw[dashed] (0,0) -- (2,-0.73);
        % A et P sur le bras
        \draw (0,0) node[left] {$A$};
        \draw (2,-0.73) node[below] {$P$};
        % Force
        \draw[->, red, thick] (2, -0.73) -- (3, 0);
        \draw (3, 0) node[text=red, right] {$\overrightarrow{F}$};
        
        % Moment
        \draw[->, blue, thick] (0, 0) -- (0, 1);
        \draw (0, 1) node[text=blue, left] {$\overrightarrow{M_A}(\overrightarrow{F})$};
    \end{tikzpicture}

    \begin{tikzpicture}
        % Axes
        \draw[->] (0,-2) -- (0,2);
        \draw[dashed] (0.5, -1.83) -- (4,0.73);
        \draw[dashed] (0,0) -- (2, -0.73);
        
        \draw [dashed] (0,0) -- (2.5,-0.37);
        \draw (0,0) node[left] {$H$};
        \draw (2, -0.73) node[below] {$H'$};
        \draw (1, -0.73/2) node[below] {$d$};

        \draw [red, thick, densely dashed, ->] (2.5,-0.37) -- ++(1, 0.73);
        \draw [red, thick, densely dashed, ->] (2.5,-0.37) -- ++(0, 1.73);
        \draw [red, thick, ->] (2.5,-0.37) -- ++(1, 2.36);
        \draw [red, dotted] (2.5, 1.3) -- ++(1, 0.73);
        \draw [red, dotted] (3.5,0.37) -- ++(0, 1.73);

        \draw (2.5,-0.37) node[below] {$P$};
        \draw (2.3, 1.3) node[text=red, left] {$\overrightarrow{F_\parallel}$};
        \draw (3.7,0.3) node[text=red, below] {$\overrightarrow{F_\perp}$};
        \draw (3.5, 2.1) node[text=red, right] {$\overrightarrow{F}$};

    \end{tikzpicture}

\end{multicols}

\begin{multicols}{2}
    \subsection*{Moment d'une force par rapport à un point}
    \[\boxed{\overrightarrow{\mathcal{M}_A}(\overrightarrow{F}) = \overrightarrow{AP} \wedge \overrightarrow{F}}\]

    \subsection*{Moment d'une force par rapport à un axe}
    \[\boxed{\mathcal{M}_\Delta(\overrightarrow{F}) = \overrightarrow{\mathcal{M}_A}(\overrightarrow{F})\cdot \overrightarrow{u_\Delta}}\]
    \[\boxed{\mathcal{M}_\Delta(\overrightarrow{F}) = \pm d \times F_\perp}\]
    
\end{multicols}
\begin{multicols}{2}
    \subsection*{Moment résultant par rapport à un point}
    \[\boxed{\overrightarrow{\mathcal{M}_A}(\overrightarrow{F}) = \sum_{i=1}^n \overrightarrow{\mathcal{M}_A}(\overrightarrow{f}_i) = \sum_{i=1}^n \overrightarrow{AP_i} \wedge \overrightarrow{f}_i}\]
    
    \subsection*{Moment résultant par rapport à un axe}
    \[\boxed{\mathcal{M}_\Delta(\overrightarrow{F}) = \sum_{i=1}^n \mathcal{M}_\Delta(\overrightarrow{f}_i) = \sum_{i=1}^n \overrightarrow{AP_i} \wedge \overrightarrow{f_i}\cdot\overrightarrow{u_\Delta}}\]
\end{multicols}

\vspace{0.3cm}

\begin{multicols}{2}
    
    \subsection*{Couple d'un système de forces}
    \[\boxed{(F_i) \text{ est un couple } \iff
    \begin{split}
        \begin{cases}
            &\sum(\overrightarrow{F_i}) = 0 \\
            \overrightarrow{\mathcal{C}} = &\sum(\overrightarrow{\mathcal{M}_A}(\overrightarrow{F_i})) \neq 0
        \end{cases}
    \end{split}}
    \]
    
    \subsection*{Moment d'inertie d'un solide}
    \[\boxed{L_\Delta = J_\Delta \omega} \text{ avec }\overrightarrow{\Omega} = \omega\overrightarrow{u_\Delta}\]
    \[\boxed{J_\Delta = \sum_{i=1}^n m_i {r_i}^2}\]
\end{multicols}

\newpage

\section*{Moment cinétique et moment d'inertie}

\begin{multicols}{2}
    \subsection*{Moment cinétique par rapport à un point}
    \[\boxed{\overrightarrow{L_A}(M) = \overrightarrow{AP} \wedge m\overrightarrow{v}}\]

    \subsection*{Moment cinétique par rapport à un axe}
    \[\boxed{L_\Delta(M) = \overrightarrow{L_A}(\overrightarrow{v})\cdot \overrightarrow{u_\Delta}}\]

\end{multicols}

\begin{multicols}{2}
    \subsection*{Moment cinétique d'un solide par rapport à un point}
    \[\boxed{\overrightarrow{L_A}(\mathcal{S}) = \sum_{i=1}^n \overrightarrow{L_A}(\overrightarrow{M_i}) = \sum_{i=1}^n m_i \overrightarrow{AM_i} \wedge \overrightarrow{v}(M_i)}\]

    \subsection*{Moment cinétique d'un solide par rapport à un axe}
    \[\boxed{L_\Delta(\mathcal{S}) = \sum_{i=1}^n L_\Delta(\overrightarrow{M_i}) = \sum_{i=1}^n m_i {r_i}^2 \dot{\theta_i}}\]
\end{multicols}

\vspace{0.3cm}

\subsection*{Moments d'inertie usuels}
\begin{multicols}{3}
    \begin{tikzpicture}
        \draw [dashed, *-*] (-1.5,0) -- (1.5,0);
        \draw [->] (0,-0.75) -- (0,0.75);
        \draw (0,0.75) node[above] {$\overrightarrow{u_\Delta}$};
        \draw (-1.5,0) node[above] {$m/2$};
        \draw (1.5,0) node[above] {$m/2$};
        \draw [<->] (-1.5,-0.2) -- (0,-0.2);
        \draw [<->] (0,-0.2) -- (1.5,-0.2);
        \draw (-0.75,-0.2) node[below] {$d$};
        \draw (0.75,-0.2) node[below] {$d$};

        \draw (0, -1) node[below] {$J_\Delta = md^2$};
    \end{tikzpicture}

    \begin{tikzpicture}
        \draw [dashed, *-*] (-0.1,0) -- (1.5,0);
        \draw [->] (0,-0.75) -- (0,0.75);
        \draw (0,0.75) node[above] {$\overrightarrow{u_\Delta}$};
        \draw (-0.1,0) node[left] {$m/2$};
        \draw (1.5,0) node[above] {$m/2$};
        \draw [<->] (0,-0.2) -- (1.5,-0.2);
        \draw (0.75,-0.2) node[below] {$d$};

        \draw (0, -1) node[below] {$J_\Delta = md^2/2$};
    \end{tikzpicture}

    \begin{tikzpicture}
        \draw [dashed, -*] (0,0) -- (1.5,0);
        \draw [->] (0,-0.75) -- (0,0.75);
        \draw (0,0.75) node[above] {$\overrightarrow{u_\Delta}$};
        \draw (1.5,0) node[above] {$m$};
        \draw [<->] (0,-0.2) -- (1.5,-0.2);
        \draw (0.75,-0.2) node[below] {$d$};

        \draw (0, -1) node[below] {$J_\Delta = md^2$};
    \end{tikzpicture}
\end{multicols}

\begin{multicols}{3}
    \subsection*{Parralélépipède}
    \begin{tikzpicture}
        % A parrallellogram
        % Fill in light gray
        \filldraw [gray!20] (0,0) -- (3,0) -- (3,1.5) -- (0,1.5) -- cycle;
        \filldraw [gray!20] (0,1.5) -- (0.2,1.7) -- (3.2,1.7) -- (3,1.5) -- cycle;
        \filldraw [gray!20] (3,0) -- (3.2,0.2) -- (3.2,1.7) -- (3,1.5) -- cycle;
        \draw (0,0) -- (3,0) -- (3,1.5) -- (0,1.5) -- cycle;
        \draw [<->] (0,-0.2) -- (3,-0.2);
        \draw [<->] (-0.2,0) -- (-0.2,1.5);
        \draw [<->] (-0.2,1.5) -- ++(0.3,0.3);
        \draw (3,0) -- (3.2,0.2);
        \draw (3,1.5) -- (3.2,1.7);
        \draw (3.2,1.7) -- (3.2,0.2);
        \draw (0,1.5) -- (0.2,1.7);
        \draw (0.2,1.7) -- (3.2,1.7);

        \draw (1.5,-0.2) node[below] {$2b$};
        \draw (-0.2,0.75) node[left] {$2a$};
        \draw (0, 1.8) node[left] {$2c$};

        \draw [densely dashed] (-0.1,0.85) -- (3.1,0.85);
        \draw [->] (3.1,0.85) -- (3.7,0.85);
        \draw (3.7,0.85) node[above] {$\overrightarrow{u_{\Delta_x}}$};

        \draw [densely dashed] (1.5,-0.1) -- (1.5,1.6);
        \draw [->] (1.5,1.6) -- (1.5,2.2);
        \draw (1.5,2.2) node[left] {$\overrightarrow{u_{\Delta_y}}$};

        \draw (1.5, -1) node[below] {$J_{\Delta_x} = m(a^2+b^2)/3$};
        \draw (1.5, -1.5) node[below] {$J_{\Delta_y} = m(a^2+b^2)/6$};
    \end{tikzpicture}

    \subsection*{Cylindre plein}
    \begin{tikzpicture}
        % A cylinder
        % Fill in light gray
        \filldraw [gray!20] (0,-0.75) -- (3,-0.75) arc (90:-90:0.5 and -0.75) -- (0,0.75) arc (90:-90:-0.5 and 0.75) -- cycle;
        \draw (3,0) ellipse (0.5 and 0.75);
        \draw (0,0.75) -- (3,0.75);
        \draw (0,-0.75) -- (3,-0.75);
        \draw (0,-0.75) arc (90:-90:-0.5 and -0.75);
        \draw [dashed] (0,0.75) arc (90:-90:0.5 and 0.75);

        \draw [densely dashed] (1.5, -0.85) -- (1.5, 0.85);
        \draw [->] (1.5, 0.85) -- (1.5, 1.4);
        \draw (1.5,1.4) node[left] {$\overrightarrow{u_{\Delta_r}}$};

        \draw [densely dashed] (0, 0) -- (3, 0);
        \draw [->] (3, 0) -- (3.8, 0);
        \draw (3.8, 0) node[above] {$\overrightarrow{u_{\Delta}}$};

        \draw [<->] (0,-0.97) -- (3,-0.97);
        \draw (1.5,-0.97) node[below] {$2l$};
        \draw (0,0) -- (-0.31,-0.58);
        \draw (-0.16,-0.4) node[right] {$r$};

        \draw (1.5, -2) node[below] {$J_{\Delta} = m(r^2)/2$};
    \end{tikzpicture}

    \subsection*{Cylindre creux}
    \begin{tikzpicture}
        % A cylinder
        % Fill in light gray
        \filldraw [gray!20] (0,-0.75) -- (3,-0.75) arc (90:-90:-0.5 and -0.75) -- (0,0.75) arc (90:-90:-0.5 and 0.75) -- cycle;
        \draw (3,0) ellipse (0.5 and 0.75);
        \draw (0,0.75) -- (3,0.75);
        \draw (0,-0.75) -- (3,-0.75);
        \draw (0,-0.75) arc (90:-90:-0.5 and -0.75);
        \draw [dashed] (0,0.75) arc (90:-90:0.5 and 0.75);

        \draw [densely dashed] (1.5, -0.85) -- (1.5, 0.85);
        \draw [->] (1.5, 0.85) -- (1.5, 1.4);
        \draw (1.5,1.4) node[left] {$\overrightarrow{u_{\Delta_r}}$};

        \draw [densely dashed] (0, 0) -- (2.7, 0);
        \draw [->] (2.7, 0) -- (3.8, 0);
        \draw (3.8, 0) node[above] {$\overrightarrow{u_{\Delta}}$};

        \draw [<->] (0,-0.97) -- (3,-0.97);
        \draw (1.5,-0.97) node[below] {$2l$};
        \draw (0,0) -- (-0.31,-0.58);
        \draw (-0.16,-0.4) node[right] {$r$};

        \draw (1.5, -2) node[below] {$J_{\Delta} = m(r^2)$};
    \end{tikzpicture}
\end{multicols}

\vspace{0.3cm}

\begin{multicols}{2}
    \subsection*{Théorème du moment cinétique par rapport à un point}
    \[\boxed{\frac{\mathrm{d}}{\mathrm{d}t}\overrightarrow{L}_A = \overrightarrow{\mathcal{M}_A}}\]

    \subsection*{Théorème du moment cinétique par rapport à un axe}
    \[\boxed{\frac{\mathrm{d}}{\mathrm{d}t}{L}_\Delta = {\mathcal{M}}_\Delta}\]
\end{multicols}

\begin{multicols}{2}
    \subsection*{Théorème du moment cinétique par rapport à un point pour un solide}
    \[\boxed{\frac{\mathrm{d}}{\mathrm{d}t}\overrightarrow{L}_A(\mathcal{S}) = \overrightarrow{\mathcal{M}_{\text{ext}/A}}}\]

    \subsection*{Théorème du moment cinétique par rapport à un axe pour un solide}
    \[\boxed{\frac{\mathrm{d}}{\mathrm{d}t}{L}_\Delta(\mathcal{S}) = J_\Delta\frac{\mathrm{d}\omega}{\mathrm{d}t} = {\mathcal{M}_{\text{ext}/\Delta}}}\]
\end{multicols}

\newpage

\section*{Energétique}

\subsection*{Energie cinétique d'un solide en rotation}
\[\boxed{\mathcal{E}_c = \frac{1}{2}J_\Delta\omega^2}\]

\vspace{0.3cm}

\begin{multicols}{2}
    \subsection*{Puissance d'une force}
    \[\boxed{P(\overrightarrow{F}) = \mathcal{M}_\Delta(\overrightarrow{F}).\omega}\]

    \subsection*{Travail d'une force}
    \[\boxed{W_{M_1 \rightarrow M_2}(\overrightarrow{F}) = \int_{M_1}^{M_2} \delta W(\overrightarrow{F}) = \int_{t_1}^{t_2} P(\overrightarrow{F})\mathrm{d}t = \int_{\theta_1}^{\theta_2}\mathcal{M}_\Delta(\overrightarrow{F})\mathrm{d}\theta}\]
\end{multicols}

\vspace{0.3cm}

\begin{multicols}{2}
    \subsection*{Théorème de la puissance cinétique pour un solide en rotation}
    \[\boxed{\frac{\mathrm{d}}{\mathrm{d}r}\mathcal{E}_c(\mathcal{S}) = \mathcal{M}_{\text{ext}/\Delta}.\omega}\]

    \subsection*{Théorème de l'énergie cinétique pour un solide en rotation}
    \[\boxed{\Delta \mathcal{E}_c(\mathcal{S}) = \int_{\theta_1}^{\theta_2} \mathcal{M}_{\text{ext}/\Delta}.\omega\mathrm{d}\theta}\]
\end{multicols}

\newpage

\section*{Frottements solides}

\[\boxed{\overrightarrow{F} = \overrightarrow{N} + \overrightarrow{T}}\]

\begin{multicols}{3}
    \subsection*{Frottement statique}
    \[\boxed{T < f_s N}\]

    
    En général
    \[\boxed{f_d \leq f_s}\]

    \subsection*{Frottement dynamique}
    \[\boxed{T = f_d N}\]

\end{multicols}

\subsection*{Lois d'Amontons-Coulomb}
Le solide ne glisse pas sur le support tant que $T < f_s N$.\\
Lors du glissement, $T = f_d N$, $\overrightarrow{T} \propto (-\overrightarrow{v}_{1/2\text{ glissement}})$.

\begin{multicols}{2}
    \subsection*{Visualisation par un cône de frottement}
    On défini $\boxed{\tan(\varphi) = f_s}$\\
    Alors il n'y aura pas de frottements tant que $\overrightarrow{R}$ est dans le cône de frottement.
    
    \subsection*{Plan incliné}
    Pour un solide posé sans vitesse : $\tan(\alpha_\text{lim}) = f_s$

\end{multicols}

\subsection*{Puissance des force de frottements}
\[P_{frot} = \overrightarrow{R}\cdot\overrightarrow{v_g} = (\overrightarrow{N} + \overrightarrow{T}\cdot\overrightarrow{v_g}) = \boxed {\overrightarrow{T}\cdot\overrightarrow{v_g}}\]

\subsection*{Puissance des actions de contact}
Lorsque deux solides $\mathcal{S}_1$ et $\mathcal{S}2$ sont en contact, la puissance des actions de contact est une puissance intérieure au système complet $\mathcal{S}_1 \cup \mathcal{S}_2$.\\ La puissance totale développée par ces actions s’exprime dans le
référentiel du laboratoire :
\begin{align*}
    P_\text{contacts} &= P_1 + P_2\\
&= \overrightarrow{R_{2/1}}\cdot\overrightarrow{v_1} + \overrightarrow{R_{1/2}}\cdot\overrightarrow{v_2}\\
&= \overrightarrow{R_{2/1}}(\overrightarrow{v_1} - \overrightarrow{v_2})\\
&= \overrightarrow{R_{2/1}}\cdot\overrightarrow{v_{g,1/2}}\\
&= \overrightarrow{T_{2/1}}\cdot\overrightarrow{v_{g,1/2}}
\end{align*}

Si pas de glissement, $\overrightarrow{v_{g,1/2}} = 0$ donc $P_\text{contacts} = 0$\\
Si glissement, $\overrightarrow{T} = -K\overrightarrow{v_{g,1/2}}$ donc $P_\text{contacts} < 0$

\end{document}