The radial force \(F_{\mathrm{r}}\) is related to a potential \(V(r)\) by $$F_{\mathrm{r}}=-\mathrm{d} V(r) / \mathrm{d} r$$ Show that the "centrifugal" force can be derived from the term \(J^{2} / 2 m r^{2}\) in the effective potential.

In physics, the radial force is a force that acts along the radius of a circular path. If an object is revolving around a point, the radial force acts either toward or away from this center point.
Mathematically, the radial force is expressed through a potential energy function, denoted as: \[-F_{\text{r}} = -\frac{dV(r)}{dr}\]
This means that the radial force is the negative gradient of the potential energy function. It essentially shows how much the potential energy changes with respect to the radial distance.
Here, the potential energy is a function of the distance, which makes it easier to analyze forces in a central force field, like gravity or electrostatics.

Potential energy is the stored energy of a system based on its position or state. In the context of radial force, the potential energy function, \(V(r)\), depends on the distance from the center.
When we mention a potential energy function, it tells us how the potential energy varies with distance.
For example:

• Gravitational Potential Energy: which depends on the height in a gravitational field.
• Elastic Potential Energy: which depends on the compression or elongation of an elastic object.

In our specific problem, we're dealing with central forces, so the potential energy is a function of \(r\), or the radial distance. By understanding how \(V(r)\) changes with \(r\), we can derive the forces acting in the system.

In many physical systems, it's useful to consider an 'effective potential'. This combines the actual potential energy with additional terms that arise from the motion of the system.
For example, in rotational systems, this often includes the centrifugal term.
The effective potential gives us more comprehensive insight into the forces acting on an object in motion. For this exercise, the effective potential, \(V_{\text{eff}}(r)\), is: \[V_{\text{eff}}(r) = V(r) + \frac{J^2}{2mr^2}\]
Here, \(V(r)\) is the central potential, and \(\frac{J^2}{2mr^2}\) is the term added for centrifugal effects.

The centrifugal potential arises in rotating systems. It's essentially a 'fictitious' potential energy term that accounts for the centrifugal force experienced by a body in a rotating frame of reference.
In our derived relation, the centrifugal potential is expressed as: \[V_{\text{cent}}(r) = \frac{J^2}{2mr^2}\]
This term is added to the effective potential to account for the additional force that mimics an outward push on the object.
By differentiating this term with respect to \(r\), we can derive the centrifugal force: \[F_{\text{cent}} = -\frac{d}{dr}\bigg(\frac{J^2}{2mr^2}\bigg) = \frac{J^2}{mr^3}\]
This centrifugal force acts outward, away from the center, and balances with the centripetal force to maintain an object's circular motion.