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

Central Forces Notes Section 8

  • After students have found $\vec{v}$ in polar coordinates (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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