Perturbed circular orbit under central force motion?

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Discussion Overview

The discussion revolves around a specific example problem from Kleppner and Kolenkow's "An Introduction to Mechanics," focusing on the derivation of an equation related to perturbed circular orbits under central force motion. Participants seek clarification on the steps involved in reaching the equation, particularly regarding the effective potential energy.

Discussion Character

  • Technical explanation
  • Homework-related

Main Points Raised

  • One participant expresses confusion about the derivation of an equation in the example problem and requests clarification.
  • Another participant suggests quoting the relevant equation or providing a link for context.
  • A participant shares a photograph of the relevant page from the textbook to aid understanding.
  • Some participants propose that differentiating the effective potential energy twice and evaluating it at a specific radius could lead to the desired result, relating it to a constant associated with the problem.
  • One participant reflects on their earlier mistake of not substituting a value in their calculations and expresses relief upon resolving their confusion.
  • A later reply offers reassurance, indicating that such moments of confusion are common in learning.

Areas of Agreement / Disagreement

Participants generally agree on the steps to take in the derivation, but there is no consensus on the clarity of the initial explanation or the specific details of the calculations.

Contextual Notes

Some limitations include potential missing assumptions in the derivation process and the dependence on specific definitions related to the effective potential energy.

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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.
 
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I suggest you quote it or give a link.
 
Here i have attached the photo graph of that page.
mathman said:
I suggest you quote it or give a link.
 

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

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