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=====Kepler's 2nd Law in Polar Coordinates Lecture (5 minutes)=====

{{courses:lecture:cflec:central_forces_notes.pdf|Central Forces Notes}} Section 8

  * After students have found $\vec{v}$ in polar coordinates ([[courses:activities:cfact:cfvpolar|Velocity and Acceleration in Polar Coordinates]]), derive an expression for $\vec{L}$ in polar coordinates (if one or more groups has done this as a part of the previous activity, have them to present their solution).
\begin{align*}
\vec{L}&=\vec{r}\times\vec{p}\\
&=\vec{r}\times\mu\vec{v}\\
&=r\hat{r}\times\mu\left(\dot{r}\hat{r}+r\dot{\phi}\hat{\phi}\right)\\
&=\mu r^2\dot{\phi}\;\hat{r}\times\hat{\phi}\\
&=\mu r^2\dot{\phi}\hat{z}\text{ (cylindrical)}\\
&=-\mu r^2\dot{\phi}\hat{\theta}\text{ (spherical)}
\end{align*}
  * Since angular momentum is conserved for a central force,
$$|\vec{L}|=\ell=\mu r^2\dot{\phi}.$$
  * This is equivalent to Kepler's 2nd Law (or the Law of Equal Areas): the line joining a planet and the Sun sweeps out equal areas during equal intervals of time.


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