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Simple Harmonic Oscillator behaviour when a potential term is added

  1. Apr 17, 2017 #1
    1. The problem statement, all variables and given/known data

    A simple harmonic oscillator has a potential energy V=1/2 kx^2. An additional potential term V = ax is added then,
    a) It is SHM with decreased frequency around a shifted equilibrium
    b) Motion is no longer SHM
    c)It is SHM with decreased frequency around a shifted equilibrium
    d) It is SHM with same frequency around a shifted equilibrium
    e)It is SHM with increased frequency around origin
    2. Relevant equations
    $$w= \sqrt {k/m}$$
    $$x = A \sin{wt}$$


    3. The attempt at a solution
    $$V = 1/2 kx^2+ax$$
    At x=0
    $$V=0$$
    Maximum potential
    $$dV/dx = kx + a$$
    I do not know if this is correct or if it is so then i do not know how to go further than this.
    Please help.
     
  2. jcsd
  3. Apr 17, 2017 #2

    PeroK

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    Perhaps think about what a "shifted equilibrium" might mean? Would a change of coordinate help?
     
  4. Apr 17, 2017 #3
    I did not understand what you meant by change of coordinate.
     
  5. Apr 17, 2017 #4

    PeroK

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    A change of coordinate is where you use another coordinate instead of ##x##. That could be anything. But in this case, you could try using a new coordinate ##y## with ##y = x + c##, where ##c## is a constant.
     
  6. Apr 17, 2017 #5
    Do you mean to suggest that I should write it as $$ V = \frac {1}{2}k(y-c)^2$$?
     
  7. Apr 17, 2017 #6

    PeroK

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    Yes, except that's not the potential you are supposed to be working with. It's:

    ##V(x) = \frac12 kx^2 + ax##
     
  8. Apr 17, 2017 #7
    Yes so then it becomes $$ V = \frac {1}{2}k(y-c)^2+a(y-c)$$
    If i expand it, $$V = \frac{1}{2}(ky^2+kc^2)-yc+ay-ac$$
     
  9. Apr 17, 2017 #8

    PeroK

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    Does that give you any ideas about a suitable value for ##c##? Hint: look at the terms in ##y##.

    Note, however, that your algebra has gone wrong.
     
  10. Apr 17, 2017 #9
    The expansion is $$V = \frac{1}{2}(ky^2+kc^2)-kyc+ay-ac$$
    Suitable value to cancel what term?
    if $$ c= \frac {a}{k}$$ I would get $$V = \frac{1}{2}(ky^2+\frac {a^2}{k})-\frac {a^2}{k}$$

    $$V = \frac{1}{2}(ky^2)-\frac {a^2}{k}$$
    But it does not look familiar.
     
  11. Apr 17, 2017 #10

    PeroK

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    What effect does a constant term have on a potential?
     
  12. Apr 17, 2017 #11
    A shift from the normal potential. So how can i infer frequency from this?
     
  13. Apr 17, 2017 #12

    PeroK

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    Adding a constant to a potential makes no effective difference. The force is the derivative of the potential, so the constant term has no effect.
     
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