5th semester files

5th-Semester-Fall-2023/ME-For-ECEs/EquationSheet/ME-Equation-Sheet.tex

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+\documentclass{IEEEtran}
+\usepackage{amsmath}
+
+\title{ME for ECEs Equation Sheet}
+\begin{document}
+\maketitle
+
+\section{Vectors}
+$$\vec{v}_R = \sum v_x \hat{x} + \sum v_y \hat{y} + \sum v_z \hat{z}$$
+
+$$\|\vec{v}\| = \sqrt{v_x^2 + v_y^2 + v_z^2}$$
+
+$$\begin{align}
+    \vec{u} \cdot \vec{v} &= \|\vec{u}\| \|\vec{v}\| \cos(\theta) \\
+    &= (u_x v_x) + (u_y v_y) + (u_z v_z)
+\end{align}$$
+
+$$\begin{align}
+    \vec{u} \times \vec{v} &= 
+    \begin{vmatrix}
+        \hat{x} & \hat{y} & \hat{z} \\
+        u_x & u_y & u_z \\
+        v_x & v_y & v_z
+    \end{vmatrix} \\
+    &= (u_y v_z - u_z v_y)\hat{x} - (u_x v_z - u_z v_x)\hat{y} + (u_x v_y - u_y v_x)\hat{z}
+\end{align}$$
+
+\section{Statics}
+$$\vec{F} = m \vec{a}$$
+$$\vec{M} = \vec{r} \times \vec{F}$$
+$$\vec{r} = \vec{s}_f - \vec{s}_0$$
+$$\sum F = 0$$
+$$\sum M = 0$$
+
+\section{Dynamics}
+
+\subsection{Kinematic Equations}
+$$\vec{s}_f - \vec{s}_0 = \vec{v}_0 t + {1\over2} \vec{a} t^2$$
+$$\vec{v}_f^2 = \vec{v}_0^2 + 2 \vec{a} \cdot (\vec{s}_f - \vec{s}_0)$$
+$$\vec{v}_f = \vec{v}_0 + \vec{a}t$$
+$$\vec{s}_f = \vec{s}_0 + \vec{v}t$$
+
+\subsection{Projectile Motion}
+$$\vec{g} = -9.81\hat{y}\left[{\text{m} \over \text{s}^2}\right] = -32.2\hat{y} \left[{ \text{ft} \over \text{s}^2}\right]$$
+Without air resistance:
+$$v_{x_0} = v_{x_f}$$
+
+
+\section{Heat Transfer}
+
+\subsection{Conduction}
+$$q_x'' = k {dT \over dx}$$
+$${dT \over dx} = {T_2 - T_1 \over L}$$
+$$q_x = q_x'' A$$
+Where:
+\begin{itemize}
+    \item[$q_x''$] is heat flux
+    \item[$q_x$] is heat rate
+    \item[$T$] is temperature
+    \item[$L$] is length
+    \item[$A$] is the contact area
+\end{itemize}
+
+\subsection{Convection}
+$$q'' = hA(T_s - T_\infty)$$
+Where:
+\begin{itemize}
+    \item[$h$] is the heat transfer coefficient
+    \item[$A$] is the contact area between the surface and the fluid
+    \item[$T_s$] is the surface temperature
+    \item[$T_\infty$] is the fluid temperature very far away from the surface
+\end{itemize}
+
+\subsection{Radiation}
+$$q_\text{ideal}'' = \sigma T_s^4$$
+$$q_\text{real}'' = \varepsilon \sigma T_s^4$$
+Where:
+\begin{itemize}
+    \item[$T_s$] is the absolute temperature
+    \item[$\sigma$] is the Stefan-Boltzmann constant
+    \item[$\varepsilon$] is the emissivity
+\end{itemize}
+
+\section{Fluids}
+\noindent
+Pascal's Law:
+$${F_1 \over A_1} = {F_2 \over A_2}$$
+Density:
+$$\rho = {m \over v}$$
+Specific Weight:
+$$\gamma = {mg \over v} = \rho g$$
+Pressure in a fluid:
+$$P_2 = P_1 + \rho g z$$
+Bernoulli Equation:
+$$p_1 + {1\over2}\rho v_1^2 + \rho g h_1 = p_2 + {1\over2}\rho v_2^2 + \rho g h_2$$
+Fluid Velocity Between Plates:
+$${U \over b} = {u \over y}$$
+Where:
+\begin{itemize}
+    \item[$U$] is the velocity of the moving plate
+    \item[$b$] is the distance between the plates
+    \item[$u$] is the velocity of the fluid at between the plates at some distance above the stationary plate
+    \item[$y$] is the distance above the stationary plate.
+\end{itemize}
+Continuity Equation
+$$A_1 v_1 \Delta t = A_2 v_2 \Delta t$$
+$$\dot{m} = \rho_1 A_1 v_1 = \rho_2 A_2 v_2$$
+
+\section{Gears}
+$$N = P d$$
+$$N = {d \over m}$$
+$$c = {d_1 + d_2 \over 2} = {N_1 + N_2 \over 2 P} = {(N_1 + N_2) m \over 2}$$
+Where:
+\begin{itemize}
+    \item[$N$] is the number of teeth
+    \item[$d$] is the pitch diameter
+    \item[$c$] is the center distance
+    \item[$P$] is the diametral pitch (Customary)
+    \item[$m$] is the module (SI)
+\end{itemize}
+\noindent
+Gear Ratio:
+$$R = {T_2 \over T_1} = {N_2 \over N_1} = {d_2 \over d_1} = {\omega_1 \over \omega_2}$$
+$$\omega = {\pi \over 30}\text{RPM}$$
+
+\noindent
+Big Gear to Small Gear:
+\begin{itemize}
+    \item Speed increases
+    \item Torque decreases
+\end{itemize}
+
+\noindent
+Small Gear to Big Gear:
+\begin{itemize}
+    \item Speed decreases
+    \item Torque increases
+\end{itemize}
+
+\noindent
+Power:
+$$P = \omega T$$
+If no power losses:
+$$P_\text{in} = P_\text{out}$$
+$$\omega_1 T_1 = \omega_2 T_2$$
+\end{document}