Examples of central forces on the path of orbit? In solving a problem from Goldstein (3.13), I solved for multiple properties of a circular orbit with the attractive central force where the path of orbit crosses the point of the force (at origin). 
The solutions were simple enough to find, but what's been in the back of my mind is what type of physical system does this represent? I am used to Kepler type problems where the central force is located within the orbit and not on it. What system would this be applied to? Or is it merely an exercise?
 A: Are you asking purely about an orbit system? Or a general system? One I can think of is simple harmonic motion. Take a mass on a spring. The force will always act towards the origin, and the mass will go through the origin. Although the definition of a central force is one that always acts towards a fixed point (in this case, the origin), I'm not 100% sure if simple harmonic motion counts as a central force. Perhaps someone can correct me here if necessary. 
I'm struggling to think of an orbit example in space which fits your problem however.
A: The orbit here can be taken as the limit as $r_0 \to a$ of the case where the orbit is an eccentric circle with radius $a$ and center a distance $r_0$ from the origin.  I solved for the potential and the force law for this general case in this answer.  In the limit of $r_0 \to a$ the results simply become
$$
U(r) = -\frac{ k}{r^4}, \quad F(r) = - \frac{4k}{r^5} \hat{r}.
$$
So this orbit could arise if the force law was proportional to $r^{-5}$ rather than $r^{-2}$ as in the Kepler problem.  There are no known two-body forces with this behavior, but one could contrive such a force law by imagining a mass (or charge) distribution spread out through some region of space acting on a massive (or charged) test particle.  Alternately, one would expect a $r^{-5}$ dependence for gravity in a Universe with 6 spatial dimensions.  (Whether this seems more or less contrived than the previous example is a matter of taste.)
As noted in my previous answer, only particles with special initial conditions (namely $L^2/\mu = k/(2a^2)$) will actually describe these clean circular paths.  The general paths for orbits in this potential will be much more complicated.  In fact, for a general $r^{-6}$ potential, most orbits will either crash into the origin (as these do) or fly off to infinity.
A: Consider this scenario in which a spring is connected to a bead and the other end of spring is connected to a circular frame and bead is set to contained on that circular frame and end of spring connected to circular frame is glued and take it as origin. 
HOPE THIS HELPS.
Note this problem is merely an excersice problem and this spring system is just to give you feel. But the type of force which attracts depends on question so do not apply mecanics of spring on problem just do what is stated there are infinite systems possible for this scenario.
