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Section 14.2 Effective Potential

The effective potential is the sum of two terms: the ordinary potential, \(-\frac{k}{r}\) for spherically symmetric gravitational or electrostatic forces; and the angular part of the kinetic energy, \(-\frac{\ell^2}{2\mu r^2}\text{.}\)

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 \(\mu\text{.}\) (Note that the dependence of \(k\) on \(\mu\) for the case of gravitational forces has been ignored.)

Activity 14.2.

Explore how the shape of the effective potential (shown in black) depends on the parameters \(\ell\text{,}\) \(k\text{,}\) and \(\mu\text{.}\)

Figure 14.2. The effective potential, shown in black is the sum of two terms: the ordinary potential, shown in blue, and the angular part of the kinetic energy, shown in red. For a given effective potential, the total energy \(E\text{,}\) shown in green will determine the shape of the orbit.