\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}
