Angular momentum & Energy using Yukawa's potential

[Nigsia]

Hello there!
I was doing my Gravitation problems and I found this problem that I'm unable to solve.

Yukawa's theory for nuclear forces states that the potential energy corresponding to the attraction force produced by a proton and a neutron is:
$$U(r) = \frac{k}{r}e^{-\alpha r},\ k<0,\ \alpha > 0$$
From the expression of it's effective potential, find the module of it's angular momentum and it's energy, for which it's possible a circular movement with a radius r₀​

I've tried several things, none of them leading to something meaningful. In fact, I know that expression for effective potential is:
$$U_{ef}(r)=U(r)+\frac{L}{2r^2}$$
So I imagine I would need to find L fist in order to get the expression for U_ef, but I'm not able to remember nor find any kind of formula linking U and L. Would you please help me out?

PS: Once I know how to find L I know how to end it, since:
$$\frac{dU_{ef}}{dr} = 0 \Leftrightarrow r = r_0$$
is the expression of the energy of a circular movement with a radius r₀

Thanks in advance.

 

[Orodruin]

I think you are misunderstanding the problem task. You need to use $r_0$ to find the angular momentum and energy. Your work so far is fine and you should continue by differentiating the effective potential.

 

[Nigsia]

Okay, I have it differentiated. How do I find angular momentum? As far as I know, I only find the energy with what I've done.

 

[Orodruin]

How does the energy look like for a circular orbit?

 

[Nigsia]

Do you mean that the Potential is twice the kinetic?

 

[Orodruin]

No. I mean: How do you express the total energy for a circular orbit? There is a very simple expression.

 

[Nigsia]

I don't really know that formula. The only thing that I can think of is $$E=\frac{U}{2}$$

 

[Orodruin]

No, it is much simpler than you are thinking. What contributions are there to the total energy?

 

[Nigsia]

I really don't know. Can you give me a hint?

 

[Orodruin]

What types of energy do you know of?

 

[Nigsia]

Kinetic and potential.

 

[Orodruin]

Right, so what are their values for a circular orbit?

 

[Nigsia]

The potential is twice the kinetic, so $$\left| E\right| = \left| \frac{U}{2} \right| = \left| K \right|$$

 

[Orodruin]

The potential is twice the kinetic, so $$\left| E\right| = \left| \frac{U}{2} \right| = \left| K \right|$$

No. This is simply wrong. You have to be aware when certain theorems hold and when they do not. The answer is much simpler and does not require anything else than very basic mechanics.

 

[Nigsia]

Could you please give me another hint? I'm really struggling to get anything clear.

 

[Nigsia]

In theory $$L=(-mkr_0(1+\alpha r_0)e^{-\alpha r_0})^{1/2}$$