Perturbed circular orbit under central force motion?

[physicsxlove]

I am self studying Kleppner and Kolenkow's an Introduction to mechanics. But i have one doubt about how they got into the equation no 3 of the example problem 9.3 in Central Force Motion.
Please clarify my doubt.

 

[mathman]

I suggest you quote it or give a link.

 

[physicsxlove]

Here i have attached the photo graph of that page.

I suggest you quote it or give a link.

 

[diegzumillo]

Seems pretty straight forward. Differentiate $U_{eff}$ twice, evaluate it at $r=r_0$ and replace $l$ by the one we obtained earlier. This should equal k, which is related to beta.

 

[physicsxlove]

Seems pretty straight forward. Differentiate $U_{eff}$ twice, evaluate it at $r=r_0$ and replace $l$ by the one we obtained earlier. This should equal k, which is related to beta.

Omg. I am so dumb. I just calculated the second derivative and then used to sit brooding over what to do next without plugging the value of l from equation 1. I am feeling embarrassed.

Anyway,,, thank you. I got it. Better late than never.

 

[diegzumillo]

No problem :) This is just the peeking-over-the-shoulder effect.