Activity 21.7.
Explore how the shape of the effective potential (shown in black) depends on the parameters \(\ell\text{,}\) \(k\text{,}\) and \(\mu\text{.}\)
Section 21.15 Effective Potential
If we compare the equation for the velocity of a one-dimensional harmonic oscillator (21.13.3)
\begin{equation}
\dot{x} =
\pm\sqrt{\frac{2}{m}\bigl(E-\frac12 (x-x_0)^2\bigr)}\tag{21.15.1}
\end{equation}
from Section 21.13 with the equation for the radial coordinate (21.14.4)
\begin{align}
\dot{r} \amp =
\pm\sqrt{\frac{2}{\mu}\bigl(E-U(r)\bigr)-\frac{\ell^2}{\mu^2r^2}}\tag{21.15.2}\\
\amp =
\pm\sqrt{\frac{2}{\mu}\Bigl(E
-(U(r)+\frac{\ell^2}{2\mu r^2})\Bigr)}\tag{21.15.3}
\end{align}
from Section 21.14, we can see an intriguing possibility. If we add together the ordinary potential \(U(r)\) and the angular part of the kinetic energy, \(-\frac{\ell^2}{2\mu r^2}\text{,}\) this sum acts the same way in (21.15.3) as the potential does in (21.13.3). Then, we can make a graph analogous to Figure 21.10 and analyze the classical turning points of the motion. The sum of the ordinary potential, \(U(r)\) and the angular part of the kinetic energy, \(-\frac{\ell^2}{2\mu r^2}\) is called the effective potential.
\begin{equation}
U_\text{eff} \equiv
U(r)-\frac{\ell^2}{2\mu r^2}\text{.}\tag{21.15.4}
\end{equation}
The applet below shows you how the effective potential depends on the parameters: (magnitude of the) angular momentum \(\ell\text{,}\) strength of the force \(k\text{,}\) and reduced mass \(\mu\) for the case \(-\frac{k}{r}\) for spherically symmetric gravitational or electrostatic forces. (Note that the dependence of \(k\) on \(\mu\) for the case of gravitational forces has been ignored.)
