Work $W = \vec{F}\cdot\vec{\Delta r} = F \> \Delta r \> \cos \alpha$

Work is a scalar, $[W] = 1\,\mbox{N} \times 1\,\mbox{m} = 1\,\mbox{Joule}$

Conservative forces: $W_{\rm c.f.}$ does not depend on path, only on endpoints \[ W_{\rm c.f.}(\vec{r} \rightarrow \vec{r}\,') = - \left[ U(\vec{r}\,') - U(\vec{r}) \right] = - \Delta U(\vec{r}) \]

Potential energy $U_F$ associated with force $F$ \[ \begin{eqnarray*} F_{\rm gravity}=-mg \quad & \rightarrow & \quad U_g=mgy \\ F_{\rm elastic}=-kx \quad & \rightarrow & \quad U_e={1\over2}kx^2 \end{eqnarray*} \]

Kinetic energy of translation with velocity $\vec{V}$ \[ K_{\rm tr} = {1\over2}\>{mV^2} \] Kinetic energy of rotation with angular velocity $\omega$ \[ K_{\rm rot} = {1\over2}\>{I\omega^2} \]

Conservation of total mechanical energy \[ \Delta \left( U + K_{\rm tr} + K_{\rm rot} \right) = W_{\rm non-c.f.} \]