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Classical Mechanics - Potential Energy Function

  1. Dec 10, 2014 #1
    1. The problem statement, all variables and given/known data
    The potential energy function of a particle of mass m is V(x) = cx/(x2+a2), where c and a are positive constants.

    Qualitatively sketch V as a function of x. Find two equilibrium points: identify which is a position of stable equilibrium, and find the period of small oscillations about it.

    2. Relevant equations

    3. The attempt at a solution
    I think I'm supposed to differentiate and let it equal to zero which gave me:

    (c(x2+a2) - 2cx2) / (x2+a2)2 = 0
    cx2 + ca2 - 2cx2 = 0
    x = a

    Putting that into the function gave me: V(x) = cx2/2x2 = c/2

    I don't know if any of this is necessary but it doesn't answer the question. Also when it says sketch do I just sub value of x in? Any help would be greatly appreciated.
  2. jcsd
  3. Dec 10, 2014 #2


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    It answers part of the question, i.e., one of the equilibrium points. In order to find the other you must find the second solution to your equation (it exists as it is a second order equation in x).

    To know which is stable you must look at the second derivative to deduce which is a min and which is a max.
  4. Dec 10, 2014 #3
    Hey Orodruin thanks for the reply. How do I find the second solution?
  5. Dec 10, 2014 #4


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    You start from the equation you obtained and solve it for the general case. You have x^2 = a^2. This has two possible solutions.
  6. Dec 10, 2014 #5
    Ah yes sorry, so the other value for x is -a. But this gives the same solution as it gets squared?
  7. Dec 10, 2014 #6
    I just realised I should be subbing a in for x so I get V(x) = ±c/2a. So from this how do I tell where the stable equilbrium is? Also how do I find the period?
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