What Is the Potential Energy and Force in a Central Force Orbit?

[TimeLordo123]

Homework Statement

The problem is related with central forces. In the problem I am given the equation of the orbit of a particle subjected to a central force (with an angular momentum "l"), r=a/(phi+1)^2 (where "r" is de distance to the center of forces and "a" a constant). I am asked for the potential energy of the particle, the force applied to it and then I am asked to discuss qualitatively the characteristics of the orbit of a particle subjected to this force, depending on its energy.

Homework Equations

E=(1/2)m(dr/dt)^2 + (l^2)/(2mr^2) + U(r)
F=-dU/dr

3. The Attempt at a Solution
Getting to the potential energy of the particle and force applied to it is rather easy, and I am sure the following expressions are correct: U(r)=E-(l^2/(2mr^2))(1+(4r/a)) and F=-(l^2/(mr^2))(2/a+1/r). I think the graph I am looking for is something like this: http://l27.imgup.net/adasdad3f46.jpg . However now I don't know what levels of energy to use, and what to say about them. My guess would be that I have to simply use one positive and one negative level of energy.

Could someone help please?

 

[ehild]

I am asked for the potential energy of the particle, the force applied to it and then I am asked to discuss qualitatively the characteristics of the orbit of a particle subjected to this force, depending on its energy.

Homework Equations

E=(1/2)m(dr/dt)^2 + (l^2)/(2mr^2) + U(r)
F=-dU/dr

3. The Attempt at a Solution
Getting to the potential energy of the particle and force applied to it is rather easy, and I am sure the following expressions are correct: U(r)=E-(l^2/(2mr^2))(1+(4r/a)) and F=-(l^2/(mr^2))(2/a+1/r). I think the graph I am looking for is something like this: http://l27.imgup.net/adasdad3f46.jpg .

Your graph is the potential energy in terms of r, but you are asked about the graph of the orbit. How does it look like? What happens with the particle with increasing phi? Can it stay on a stationary orbit? What energy it should have then?