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\textbf{1.MECÁNICA DE FLUIDOS}\newline
\underline{HIDROSTÁTICA}\newline
Densidad, presión, Ec. Fundamental:
$$\rho =\displaystyle \frac{m}{V},\quad p =\frac{F}{A}, \quad \frac{\dd p}{\dd z} = -\rho g\to p = p_o+\overbrace{\rho  g h}^{\mathrm{p.hidrost.}}$$ 
Fuerza en una pared: $$F_p = \rho g a \cdot \frac12 H^2$$
Flotación:$$F_A = mg \Rightarrow \rho_l V_s g = mg$$
Fluido compresible ($H_0 = 8$)  
$$ p = p_o\cdot e^{\frac{-z}{H_0}}$$ $$\rho = \rho_0\cdot e^{\frac{-z}{H_0}} $$
Tensión superficial:
$$\sigma \ell = F_r \Rightarrow F = \Delta P \cdot S$$ $$\sigma 2\pi R\cos\theta = mg \Rightarrow h = \frac{2\sigma\cos\theta}{\rho g R}$$
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\underline{HIDRODINÁMICA}\newline
Ec. continuidad: $$v_1S_1= v_2S_2 = Q$$
Ec. Bernoulli: $$p_1 + \frac 12 \rho v_1^2 + \rho g h_1 =p_2 + \frac 12 \rho v_2^2 + \rho g h_2 = C $$
Relaciones: $$v = \sqrt{2gh}; \quad v_1 = \sqrt{\frac{2gh}{\left({\frac{S_1}{S_2}}\right)^2-1}}; \quad Q = \sqrt{\frac{2gh}{S_1^2-S_2^2}}S_1S_2  $$
Sifón: $$v_2^2 = 2g(l+d)$$
Viscosidad: $$F =\eta S\frac{v}{z};\quad R = \frac{8\eta L}{\pi r^4};\quad N_r = \frac{\rho v R}{\eta}$$
Fla. de Poiseuille: $$v = \frac{\Delta P}{4\eta \ell}(R^2-r).$$
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\textbf{2. TERMODINÁMICA}
$$T_f = \sum_i \frac{m_iT_i}{m_i};\quad \frac{c_2}{c_1} = \frac{m_1(t_1-t_f)}{m_2(t_2-t_f)};$$
$$ \Delta L = \alpha L_0\Delta T;\quad \Delta V = \beta V_0\Delta T ;$$ 
$$\Delta Q = mc_e\Delta T = c_c\Delta T;\quad  W = \frac{\Delta Q}{\Delta t}$$
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\underline{TRANSPORTE DE CALOR}\newline
\textbf{1. Conducción:}\newline
Flujo:
$$I_T = \frac{\Delta Q}{\Delta t} = \kappa A\frac{T_+ - T_-}{L};\quad j_T = -\kappa\frac{\dd T}{\dd x}$$
Conductividad térmica de una pared:
$$ j_T = \frac{\kappa}{\ell}\Delta T;\; R_f = \frac {\ell}{\kappa};\quad I_T = j_T S = \frac {\kappa S}{\ell}(T_0-T_{\ell});\quad \Delta T = I\cdot R_T$$
2 Barras en Serie: 
$$\Delta T = IR_{eq};\quad R_{eq} = \frac{\ell}{S}\left({\frac{1}{\kappa_1}+\frac{1}{\kappa_2}}\right)$$
$$I = \frac{\Delta T}{R_{eq}} = (T_0-T_2)\cdot \frac{S}{\ell}\left({\frac{\kappa_1\kappa_2}{\kappa_1 + \kappa_2}}\right)$$
2 Barras en Paralelo:
$$\Delta T_1 = \Delta T_2 = \Delta T;\quad I = I_1 + I_2;\quad \Delta T = I_i R_i$$
$$\Delta T = R_{eq} I$$
Aislamiento:
$$\ell=\sqrt{\alpha\Delta T};\quad \alpha = \frac{\kappa}{\rho_c}$$
$$c = \frac{\kappa_2}{\kappa_1}\frac{\ell_1}{\ell_2} = \sqrt{\frac{\rho_2c_2\kappa_2}{\rho_1c_2\kappa_1}}$$
$$R = \frac{d}{\kappa S}$$

\textbf{2. Convección:}\newline
Ley de enfriamiento de Newton:
$$\frac{\Delta Q}{\Delta t } = qS[T_S - T_{\infty}];$$
$$ R_{conv} = \frac{\ell}{qS};\quad R_{eq} = \sum_ i R_i$$
Flujo y Fórmula de Wien:
$$j = \kappa\frac{\Delta T}{\ell};\quad \lambda_m T = A$$
\textbf{3. Radiación:}
$$I_R = \frac{I_E}{I_A}$$
Fórmula de S. Boltzmann:
$$j_T = \frac{2\pi^5}{15}\frac{K_B^4T^4}{h^3c^2}$$
Pérdida de calor:
$$\frac{\Delta Q}{\Delta t}\left.{\!\!\frac{}{}}\right|_{\mathrm{neta}}  = e\sigma S(T^4_s -T^4_{\infty} )$$
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\underline{Gases Ideales}
$$pV = nRT;\quad k_B = \frac{R}{N_A};\quad n = \frac{m}{\mu};\quad \mu=m_0N_A$$
$$pV = Nk_B T = nN_ak_BT = nRT$$
Procesos: Isotérmicos, Isobáricos e Isocóricos:
$$pV = cte.\quad V = V_0(1+\alpha T);\quad p = p_0\alpha T$$
Cálculo cinético de la presión:
$$p = \frac 13 \rho\bar{v}^2$$
Energía cinética media:
$$\bar{E} = \frac 32 k_B T; \quad \bar{v}^2 = \frac{3k_B T}{m} \to v_{rcm} =  \sqrt{\frac{3k_BT}{m}}$$
Velocidad media at.:
$$\bar{v}_{at} = \frac{(R_2-R_2)\omega R_2}{S}$$
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\textbf{Distribución de Maxwell (Boltzman)}:
$$f(v) = 4\pi\left({\frac{m}{2\pi k_BT}} \right)^{3/2}v^2e^{\frac{-mv2}{2k_BT}} $$
Velocidad máxima, velocidad media y $v_{rcm}$:
$$v_M = \sqrt{\frac{2k_BT}{m}};\quad \bar{v} = \sqrt{\frac{8k_BT}{\pi M}};\quad v_{rcm} = \sqrt{\frac{3k_BT}{m}} $$
Relación $f(v) \to f(\varepsilon)$
$$f(\varepsilon) = \frac{2}{\sqrt{\pi}}\frac{1}{(k_BT)^{3/2}}\sqrt{\varepsilon}e^{\frac{-\varepsilon}{k_BT}}$$
Conclusión: $Q = c\mu n\Delta T$\newline
I Ley de la termodinámica:
$$Q = \Delta U + W \Rightarrow \\d Q = \dd U + p\dd V$$
$$C_{\mu}\dd T = \\d U_{\mu} + R\dd T \Rightarrow C_p =C_v + R$$
Th. de equipartición de energía i recorrido libre: 
$$c_1 = i\frac 12 k_B \Longrightarrow c_{\mu} = i\frac 12 R;\quad\lambda = \frac{1}{\sqrt 2 \pi d^2 n}$$
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\textbf{3. ONDAS}
Ecuación de un oscilador armónico (MAS):
$$\ddot x + \omega^2x = 0; \quad \omega = 2\pi\nu;\quad T = 2\pi\sqrt{\frac mk};\quad x = A\cos({\omega t +\varphi_0});$$ 
Energía:
$$E = \frac 12 mv2 + \frac 12 kx^2 \Longrightarrow m\ddot x + kx = 0$$
$$E = \frac 12 kA^2$$
Péndulo simple:
$$ m\ddot x = mg\sin\alpha;\quad x = L\alpha;\quad T = 2\pi\sqrt{\frac{L}{g}}$$
Tubo de U:
$$\Delta F = pS - \rho g S \Delta H$$
Suma de 2 M.A.S:
$$x = x_1+x_2 = 2A\cos\frac{\varphi}{2}\cos\left({ \omega t + \frac{\varphi}{2}}\right)$$
Oscilación periódica: $y(t_0+T)=y(T)$\newline
Oscilación armónica: $y(t) = A\cos(\omega t + \varphi_0)$\newline
Función de onda: $y = f(x-vt)$\newline
Relaciones:
$$\lambda = vt \to v = \lambda\nu \to \omega = kv;\quad v = \sqrt{\frac{F_T}{\mu}};\quad k = \frac{2\pi}{\lambda}$$
$$a_{max} = A\omega^2;\quad v_{max} = A\omega$$
Función de onda armónica:
$$y(x,t) = A\sin(kx\pm \omega t +\varphi_0)$$ $$v_y(x,t) = A\omega\sin(kx-\omega t)$$ $$a_y(x,t) = -A\omega\cos(kx-\omega t) = -\omega^2y(x,t)$$
Energía, potencia e intensidad de una onda:
$$E_c = \frac 12 \mu^2 v_y^2 = \frac 14 \mu A^2$$
$$E=E_c+U_p = \frac 12\mu A^2\omega^2 = \frac{1}{2}\mu v^2_{\mathrm{max}}$$ 
$$\Delta E = \frac 12 \mu A^2 \omega^2 v\Delta t$$
$$P = \frac{\Delta E}{\Delta t} = \frac 12 \mu A^2 \omega^2 v $$
$$P = \frac{1}{2}\mu v\omega^2A^2 \to \mathrm{cuerda}$$
$$P = \frac{1}{2}\rho v\omega^2A^2\to \mathrm{medio\;volum.}$$
$$I=\frac{\Delta E}{\Delta S\Delta t} = \frac{P}{\Delta S} = \frac 12 \rho v\omega^2A^2$$
Ondas estacionarias, principio de superposición:
$$y_T = y_1+y_2 = 2A\cos(kx)\cos(\omega t)$$
$$A_i + A_r = A_t;\quad A_ik_1 - A_rk_2 = A_tk_2$$
$$\alpha = \frac{v_2}{v_1} =\frac{k_1}{k_2};\quad \omega = k_1v_1 = k_2v_2;\quad \alpha = \frac{\rho v_2}{\rho v_1} $$
Desfases: (1)temoral, (2) $=y\neq t$, (3) $\neq y = t$:
$$\Delta t = \frac{\Delta x}{v};\quad \Delta\varphi = \omega\Delta t;\quad \Delta\varphi = k\Delta x$$ 
Coeficientes de transmisión, reflexión; potencia, coeficiente de reflectividad y transmitancia:
$$t = \frac{A_t}{A_i} = \frac{2\alpha}{\alpha + 1};\quad r =\frac{A_r}{A_i} = \frac{\alpha-1}{\alpha + 1}$$
$$P = \frac 12\mu v \omega^2 A^2 = \frac 12 F_T\omega^2 \frac{A^2}{v};\quad R = \frac{P_r}{P_i} = \frac{A^2r}{A^2i}=\frac{(\alpha -1)^2}{(\alpha + 1)^2}=r^2$$  
$$T = \frac{P_t}{P_i}=\frac{A_t^2/v_2}{A_i^2/v_1} = \frac{t^2}{\alpha}=\frac{4\alpha}{(1+\alpha)^2};\quad R+T=1$$
Armónicos(a) extr.fijos; (b) 1 extr. libre:
$$(a) L = n\frac{\lambda_n}{2};\quad\lambda_n = \frac{2L}{n};\quad \nu_n = n\frac{v}{2L} = n\nu_1$$
$$y_n(x,y) = 2A\sin(k_nx)\sin(\omega_n t)$$
$$(b)L = n\frac{\lambda_n}{4};\quad\lambda_n = \frac{4L}{n};\quad \nu_n = n\frac{v}{4L} = n\nu_1; n=1,3,5... $$
Ondas sonoras:
$$S(x,t)=S_n\cos(kx-\omega t);\quad \Delta p(x,t)=\Delta p_n\sin(kx-\omega t)$$
$$\Delta P_m = \rho v\omega S_m;\quad v=\sqrt{\frac{B}{\rho}} = \sqrt{\gamma\frac{RT}{\mu}}=\sqrt{\gamma\frac{k_BT}{m}}$$
$$I=\frac 12\frac{(\Delta p_m)^2}{(\rho v)^2};\quad \beta = 10\log\frac{I}{I_0}$$
Efecto Doppler (1) OMFR, (2) ORFM, (3)OMFM:
$$(1)\nu' = \nu\left({1\pm\frac{v_{ob}}{v}}\right);\quad (2)\nu' =\nu\left({\frac{1}{1\pm\frac{v_f}{v}}}\right)  $$
$$(3.1):\nu'=\nu\frac{v\pm v_{ob}}{v\mp v_f};\quad (3.2)\nu' = \nu\frac{v-v_{ob}}{v+v_f}$$
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