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Stable equilibium of a potential

  1. Feb 21, 2012 #1
    1. The problem statement, all variables and given/known data
    A particle of mass moving in one dimension has potential energy ##U(x)=U_0[2(x/a)^2-(x/a)^4]##, where #U_0# and ##a## are positive constants.
    (i)What is the angular frequency ##w## of oscillations about the point of stable equilibrium?
    (ii)What is the minimum speed the particle must have at the origin to escape to infinity?
    (iii)At ##t=0## the particle is at the origin and its velocity is positive and equal in magnitude to the escape speed of part (d). Find ##x(t)##

    3. The attempt at a solution
    I have no idea about (i), though I know the stable equilibrium point is x=0.
    My result of (ii) is ##\sqrt{\frac{-2U_0}{m}[2(x/a)^2-(x/a)^4]}##
    But, here comes the problem. The speed at x=0 is 0, so it must stay at the origin forever.
    I might get something wrong in obtaining the expression of (ii).
    Any help would be appreciated.
     
    Last edited: Feb 21, 2012
  2. jcsd
  3. Feb 21, 2012 #2

    ehild

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    Homework Helper
    Gold Member

    Plot the potential function. From that, you can understand what energy the particle needs to escape from the potential well.
    As for (i) you need to find the angular frequency of small oscillations around x/a=0. If it is small, the fourth power can be ignored with respect to (x/a)^2 and the potential energy function approximates that of a spring.
    I do not understand qestion (iii) What is x(x)? was not that v(x) instead?

    ehild
     
  4. Feb 21, 2012 #3
    The graph would be something like this ,
    the potential at maxima (x=+-a) needs to be equated to KE
    you will get velocity (min)
     

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