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h1.  Interactions

h3. General Definitions

h5. Work
{latex}\begin{large}\[ W = \int_{r_0}^{r_f} \vec{F}\cdot d\vec{r}\]\end{large}{latex}

h5. Potential Energy
{latex}

\begin{large}\[\Delta U = -W_{\rm conservative} = -\int_{A}^{B} \vec{F}_{c}\cdot d\vec{r}\]\end{large}

{latex}

h5. Power
{latex}\begin{large}\[P = \vec{F}\cdot\vec{v} \qquad \qquad P_{rot}=\frac{dW_{rot}}{dt} = \tau_{S}\omega\]\end{large}{latex}

h5. Impulse
{latex}

\begin{large}\[ I = \int_{t=0}^{t=t_f} \vec{F}(t)\:dt \]\end{large}

{latex}

h5. Torque

{latex}\begin{large}\[\vec{\tau}_{S} = \vec{r}_{PS} \times\vec{F}_{P} \qquad \qquad |\vec{\tau}_{S}| = |\vec{r}_{PS}||\vec{F}_{P}| \sin\theta = r_{\perp}F = r F_{\perp}\]\end{large}{latex}

h5. Net Torque

{latex}

\begin{large}\[ \sum_{i} \vec{\tau}_{S,i} = \vec{r}_{S,cm}\times \sum_{i} \vec{F}_{i}^{\:ext} + \sum_{i} \vec{r}_{cm,i} \times \vec{F}_{i}^{\:ext} \]\end{large}{latex}

h3. Specific Interactions

h5. Gravity -- Universal

{latex}\begin{large}\[ \vec{F}_{12} = - G\frac{m_{1}m_{2}}{r_{12}^{2}}\hat{r}_{12} \qquad\qquad U_{12}(r) = - G\frac{m_{1}m_{2}}{r_{12}}\]\end{large}{latex}

h5. Gravity -- Near Earth

{latex}

\begin{large}\[ F = mg \mbox{ (directed straight downward)} \qquad \qquad U(y) = mgy \]\end{large}

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h5. Contact Force

{latex}\begin{large}\[ \vec{F}_{contact} = \vec{N} + \vec{f}\]\end{large}{latex}

h5. Friction -- Static

{latex

}\begin{large}\[ 0 \le f_{s} \le f_{s,max} = \mu_{s}N \mbox{ (directed opposite net force neglecting friction)} \]\end{large}{latex}

h5. Friction -- Kinetic

{latex}

\begin{large}\[ f_{k} = \mu_{k}N \mbox{ (opposes motion with respect to the surface)}\]\end{large}

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h5. Springs

{latex}\begin{large}\[ F = k|\Delta x| \mbox{ (restoring)} \qquad \qquad U(x) = \frac{1}{2} k x^{2} \]\end{large}{latex}

h1. System Characteristics

h5. Center of Mass

{latex}\begin{large}\[\vec{R}_{cm} = \frac{1}{m^{total}} \sum_{i=1}^{i=N} m_{i}\vec{r}_{i} \rightarrow \frac{1}{m^{total}} \int_{body} dm\: \vec{r} \]\end{large}{latex}

h5. Velocity of Center of Mass

{latex}

\begin{large}\[\vec{V}_{cm} = \frac{1}{m^{total}} \sum_{i=1}^{i=N} m_{i}\vec{v}_{i} \rightarrow \frac{1}{m^{total}} \int_{body} dm\: \vec{v} \]\end{large}

{latex}

h5. Momentum

{latex}\begin{large}\[ \vec{p} = m\vec{v} \qquad \qquad \vec{p}^{\:sys} = \sum_{i=1}^{N} m_{i}\vec{v}_{i} \]\end{large}{latex}

h5. Translational Kinetic Energy

{latex}

\begin{large}\[ K = \frac{1}{2} mv^{2} \qquad \qquad \Delta K = \frac{1}{2}mv_{f}^{2} - \frac{1}{2}mv_{0}^{2}\]\end{large}{latex

}

h5. Period

{latex}\begin{large}\[ T = \frac{2\pi}{\omega} \]\end{large}{latex}

h5. Frequency 

{

latex}\begin{large}\[ f = \frac{1}{T} = \frac{\omega}{2\pi}\]\end{large}{latex}

h5. Moment of Inertia

{latex}\begin{large}\[ I_{S} = \int_{body} dm\:r_{\perp}^{2} \qquad\qquad I_{S}=md^{2}+ I_{cm} \mbox{ (Parallel Axis Theorem)}\]\end{large}{latex}

h5. Rotational 

Kinetic Energy

{latex}\begin{large} \[ K = \frac{1}{2}I\omega^{2}\]\end{large}

{latex}

h5. Total Kinetic Energy

{latex}\begin{large}\[ K^{total} = \frac{1}{2}m^{total}v_{cm}^{2} + \frac{1}{2}I_{cm}\omega^{2}\]\end{large}{latex}

h5. Angular Momentum 

{latex}\begin{large}\[ x(t) = A\cos(\omega t) + B\sin(\omega t)\]\end{large}{latex}

h3. Rotational Motion

{

latex}\begin{large}\[\vec{\omega} = \omega\hat{k}\qquad\qquad \omega \equiv \frac{d\theta}{dt} \qquad \qquad \alpha \equiv \frac{d\omega}{dt} \equiv \frac{d^{2}\theta}{dt^{2}} \] \[ \vec{\alpha} = \alpha \hat{k} \qquad \qquad \omega(t)-\omega_{0} = \int_{t' = 0}^{t'=t} \alpha(t')\:dt'\qquad\qquad \theta(t)-\theta_{0} =\int_{t'=0}^{t'=t}\omega(t')\:dt'\] \[\vec{a} = -R\omega^{2}\hat{r} + R\alpha\hat{\theta}\]\end{large}{latex}

h3. Mechanical Energy and 

{latex}

h3. Momentum and Impulse

{latex}\begin{large}\[ \vec{F}_{ext}^{\:total} = \frac{d\vec{p}^{\:sys}}{dt} \qquad \qquad \Delta \vec{p} = \vec{I}\]\end{large}

{latex}


h3. Angular Momentum and Impulse

{latex}\begin{large}\[ \vec{\tau}_{S} = \frac{d\vec{L}_{S}}{dt} \qquad \qquad \int_{initial}^{final} \vec{\tau}_{S,ext}dt = \vec{L}_{S,final}-\vec{L}_{S,initial} \]\end{large}{latex}