A student is invited to “act out” motion corresponding to a plot of effective potential vs. distance. The student plays the role of the “Earth” while the instructor plays the “Sun”.
A student is invited to “act out” motion corresponding to a plot of effective potential vs. distance. The student plays the role of the “Earth” while the instructor plays the “Sun”.
We typically begin this activity with a lecture about the classical equations of orbital motion, specifically a derivation that reformulates 2-D equations of orbital motion to 1-D by using the effective potential.
Just before initiating the activity, the following plot is made on the board as a reference for the student. It may be useful to contrast this plot with one for the harmonic oscillator since that case is the most familiar potential energy plot.
activities:content:photos:cfeffpotential.jpg?200
This is an 8 minute video clip of the activity activities:content:video:cfveffkin.wmv
No separate wrap-up discussion is necessary, but the take home message should be that the plot of the effective potential has buried in it information about the angular momentum, and that the corresponding energy graph only tells you about the radial motion.
It is valuable to discuss the specific parts of the conservation of energy equation \[E = {{1}\over{2}} \mu \dot{r}^2 + {{1}\over{2}} {l^2 \over \mu r^2} - {k \over r}\] explicity identifying the effective potential \({{1}\over{2}} {l^2 \over \mu r^2} - {k \over r}\), the radial kinetic energy \({{1}\over{2}} \mu \dot{r}^2\), and the \(\phi\) component of the kinetic energy \({{1}\over{2}} {l^2 \over \mu r^2}\).