A modeling approach to the 8.01 final exam equation sheet.  

||Page Contents||
|{toc:indent=10px}|

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}\[P = \vec{F}\cdot\vec{v}\]\end{large}{latex}

h5. Impulse
{latex}\begin{large}\[ I = \int_{t=0}^{t=t_f} \vec{F}(t)\:dt \]\end{large}{latex}

h5. Torque

\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}

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}{latex}

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}{latex}

h5. Springs

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