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\title{Formulari mecànica}
\author{Manel Bosch}
\date{}
\begin{document}
\maketitle
\noindent
\underline{\textbf{1. CINEMÀTICA}}\newline
\underline{Defs.}
$$\vec v = v\hat v;\quad \vec a = \vec a_t +\vec a_n = \frac{\dd v}{\dd t}\hat v + v\frac{\dd\hat v}{\dd t};\qquad \vec a_n = \frac{\vec v^{\;2}}{R}\hat u_n$$
\underline{Parab:}
$$\vec a = (0,-g);\;\; \vec v=(v_{0x},v_{0y}-gt);\;\; \vec r = (x_0+v_{0x}t,y_0+v_{0y}t-\frac 12 gt^2) $$
Abast: $y=y_0$; $y_{max} = v_{0y}(t)=0$ o $y'(x)=0$.\newline
\underline{MHS}:
$$x(t) = A\cos(\omega t);\quad a(t) = -\omega^2 \cdot x(t)$$
\underline{Curv:}
$$\dot\theta =\frac{\dd\theta}{\dd t} = \omega;\quad \ddot \theta = \frac{\dd^2\theta}{\dd t^2} = \alpha$$
\underline{Polars}
$$\vec r = r\hat u_r;\quad \vec v =\dot r\hat u_r +r\dot\theta \hat u_\theta;\quad \vec a = (\ddot r -r\dot\theta^{\;2})\hat u_r +(2\dot r\dot \theta + r\ddot \theta)\hat u_{\theta}$$
$\qquad \qquad$\underline{ Mov. circular en polars:} $$\vec r: R\hat u_r;\quad \vec v = R\dot\theta \hat u_\theta;\quad \vec a = -R\omega^2\hat u_r +R\alpha \hat u_\theta$$
\underline{Descrip. vect Mov. Circ.}:
$$\vec a_t = \vec\alpha\times\vec r;\quad \vec v = \vec\omega\times\vec r;\quad \vec a_n = \omega\times(\vec\omega\times\vec r)$$
\underline{Mov. relat. transl.}
$$\vec r\;' = \vec r -\vec R;\quad \vec v\;' = \vec v-\vec V;\quad \vec a\;' = \vec a -\vec A;\quad \vec A = 0 \to \mathrm{SRI} $$
\underline{Mov. relat. rot.}
$$\vec r = \vec r\;';\quad \vec v = \vec v\;' +\vec\omega\times\vec r;\quad \vec a = \vec a\; ' +2\vec\omega\times\vec v\;' +\vec\omega\times(\vec\omega\times\vec r) $$
\underline{Mov. rel. terra}
$$a_{cf} = \omega ^2 r\cos\lambda\;\quad \vec{a}_{co} = 2\vec\omega\times\vec v';\quad g = g_0-\omega^2R\cos^2\theta$$
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\textbf{2. DINÀMICA}\newline
\underline{Defs.}
$$\vec{F} = \frac{\dd \vec p}{\dd t};\quad \vec p \equiv m\vec v$$
$$\vec\tau = \vec r\times \vec F;\quad \vec \ell = \vec r\times\vec p;\qquad \vec \tau = \frac{\dd\vec\ell}{\dd t}$$
$$\vec I = \int_{t_0}^t \vec F(t^{'})\dd t^{'};\quad \vec I = \Delta\vec p$$
\underline{Fict.}
$$\vec F -m\vec A = m\vec a \to \vec F_{fict}= -m\vec A$$
\sep
\textbf{3. TREBALL I ENERGIA}\newline
\underline{Defs.}
$$\delta W = \vec F\cdot\dd\vec r \to W_{1\to 2}= \int_{\vec r_1}^{\vec r^2}\vec F\cdot\dd\vec r^{\;'};\quad W_{1\to 2} = \Delta E_c$$
\underline{Força conservativa}
$$\oint_{\gamma}\vec F\cdot \dd\vec r = 0;\; \vec\nabla\times \vec F = \vec 0;\quad\vec F = -\vec\nabla U;\quad W = -\Delta U; $$
$$\Delta E=\Delta(E_c+ U) = 0; \omega = \sqrt{\frac Km} = \sqrt\frac{U''(x_0)}{m}$$
\underline{Força no conservativa}
$$W_{NC} = \Delta U + \Delta E_c = \Delta E = -\Delta U_{int};\quad \Delta E + \Delta U_{int} = 0$$
\sep
\textbf{4. SISTEMES DE PARTÍCULES}\newline
\underline{Centre de masses}
$$M\vec r_{CM} = \sum_{i=1}^{N} m_i\vec r_i;\;\; \;\; M\vec r_{CM} \int_{\mathcal D} \vec r\dd m;\;\;   $$
$$ M = \sum_i m_i;\quad M = \int_{\mathcal D} \dd m $$
\underline{2 partic.}
$$\vec r_{CM} = \frac{m_1\vec r_1 + m_2\vec r_2}{m_1+m_2}\equiv R;\quad \mu = \frac{m_1m_2}{m_1+m_2};\quad \vec r = \vec r_2-\vec r_1$$
$$E_c = \frac 12 M\dot{\vec R} ^{\;2} + \frac 12 \mu\dot{\vec r}^{\;2}$$
$$\vec F_{ext}=0;\quad \vec F_{int} \neq 0 ;\quad M\ddot{\vec R} = 0;\quad \vec F = \mu\ddot{\vec{r}}$$
\underline{Energies}
$$E_c = \frac{1}{2} M \vec v_{CM}^{\;2} + \frac 12\sum_i m_i\left(\vec v_i -\vec v_{CM}\right)^2$$
\underline{Xocs}
$$\sum_i \vec F_{ext} = 0 \to \vec P = \mathrm{cte.}$$
\underline{Estudiat al CdM}:
$$\vec P_1 = -\vec P_2:\quad \vec P_1\;' = -\vec P_2\; ' \to \vec P_{CM} = \vec P_{CM}\;'$$
\begin{itemize}
\item 1D:\newline
$\to$Elàstic $(E_{c_{rel}} = \mathrm{cte.})$:$\ v_2'-v_1' = -(v_2-v_1)\;$
$$v'_1 = \frac{(m_1-m_2)v_1+2m_2v_2}{m_1+m_2};\quad v'_2 = \frac{(m_2-m_1)v_2 +2m_1v_1}{m_1+m_2}$$
$\to$ Totalment intelàstic ($v_1' = v_2' = v_{CM}$): $\\{}\qquad\qquad v_2'-v_1' = -e(v_2-v_1)$\newline
\end{itemize}
\underline{Sistemes massa variable:}\newline
\begin{align}mv = (m-\dd m)(v+\dd v)+\dd m u \nonumber \\ m\dd v = -m(-v)\dd m \nonumber\\ v(t)-v(0) = -v_{expuls}\ln\frac{m(t)}{m(0)}\nonumber \end{align}
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\textbf{5.ROTACIONS i SÒLID RÍGID}\newline
\underline{Defs.}
$$\vec \tau = \frac{\dd\vec\ell}{\dd t} \quad\mathrm{respecte \;0 \;i \;CM(SRNI)}$$
$$\vec L = \sum_i\vec\ell_i = \vec r_{CM}\times M\vec v_{CM}+\vec L_{CM}$$
\underline{Sòlid rígid}: $|\vec r_i -\vec r_j| = \mathrm{cte}\;\;\forall i,j$
\underline{Translació}:
$$\vec F_{ext} = M\cdot\frac{\dd\vec v_{Cm}}{\dd t}$$
\underline{Rotació (pura)}
$$\vec L =\sum_i\vec \ell_i= \sum_i m_i\vec r_i \times (\vec\omega\times\vec r_i) = \mathcal{I}\cdot\vec\omega$$
$$L_z = I\omega$$
\underline{Moment d'inèrcia}
$$I = \sum_i m_i\rho_i^2; \quad  I = \int\dd I = \int_{\mathcal D} \rho^2\dd m;\quad \rho_i = r_i\sin\theta_i$$
\underline{Teorema d'Steiner}
$$I = I_{CM} + Mh^2; \qquad h \equiv \mathrm{dist.\; entre\; els\; 2 \;eixos}$$
\underline{Energia cinètica}
$$E_c = \frac 12 Mv_{CM}^2+ \frac 12 I\omega^2$$
$$\frac 12 I\omega^2 = \frac 12\frac{L^2}{I}$$
\underline{Dinàmica sòlid rígid}:
$$\sum_i \vec F_i = M\vec a_{CM};\qquad \sum_i \vec \tau_i = I\vec\alpha$$
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\textbf{6. GRAVITACIÓ}\newline
\underline{Força, camp, potencial i energia potencial}
$$\vec F_{12} = -G\frac{m_1m_2}{r_{12}^2}\hat r_{12};\quad \vec F = -Gm_0\sum_{i=1}^{N}\frac{m_i}{r_{i0}^2}\hat r_{i0}$$
$$\vec g = \frac{\vec F}{m_0} = -G\sum_i \frac{m_i}{r_{i0}^2}\hat r_{i0}; \quad \vec g = -g\int_{\mathcal D}\frac{\dd m}{ r'^2} \hat r'\;\quad r' =\dd(\dd m, P)$$ 
$$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!W = -G\int_{r_{10}}^{\infty} \frac{m_1m_0}{r_{10}^2}\dd r_{10} - G\int_{r_{20}}^{\infty} \frac{m_2m_0}{r_{20}^2}\dd r_{20} -\ldots = -Gm_0\sum_{i=1}^N \frac{m_i}{r_{i0}} = U(P) $$
$$V(P)=\frac{U(P)}{m} \qquad V = -G\int_{\mathcal D}\frac{\dd m}{r'}$$
$$\vec F = -\vec\nabla U \qquad \vec g = -\vec\nabla V$$
\underline{Teorema de Gauss}
$$\phi = \oint_{\mathcal S} \vec g\cdot\dd\vec S = -4\pi G M_{int}$$
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