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Undamped Harmonic Oscillator Problem

  1. Mar 30, 2015 #1
    1. The problem statement, all variables and given/known data
    Find the initial conditions for 2 interchangeable harmonic oscillators (undamped) so that they have the same amplitude of oscillation.

    2. Relevant equations
    x(t)=Xm*cos(wt+ϕ)


    3. The attempt at a solution
    The amplitude of the function is given by Xm so I would have thought that just having the same value for Xm would be the correct answer however I am obviously overlooking something since the answer is w2*x02+v02

    w2*x02 appears to be the maximum acceleration of the function however I am at a complete loss as to how where they got v02
    Also it's possible that for X0 they are referring to Xm, my university randomly changes notation from time to time for some reason.
     
  2. jcsd
  3. Mar 31, 2015 #2

    BvU

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    Confusing, isn't it ?
    For the harmonic oscillator you have a 2nd order differential equation, so there are two initial conditions. You can guess which two quantities need to be given an initial value. What is asked of you in this exercise is that you fill in two symols, e.g. v0 and x0 and find out what the resulting amplitude of the oscillation is in terms of those two. That expression is the relationship you are looking for: if that expression yields the same value, then the amplitude comes out the same too.

    You answer is correct, but you have to express it in the initial conditions.

    Your university is not wrong this time :wink:
     
  4. Mar 31, 2015 #3
    Sorry I'm still quite lost as to how to get the correct answer. After looking through some resources it appears as though the differential equation is d^2(x)/dt^2+w2*x=0 although I cant see what to do with it
     
  5. Mar 31, 2015 #4
    Sorry for double post but I can't seem to edit my last post anymore. It appears as though this is actually a conservation of energy question. I am still getting the wrong answer after using the formula (1/2)*k*xo2+(1/2)*mv02=(1/2)*kxm2
    After I solve this for xm I obtain x02+vo2/w2=xm2 which is still the wrong answer.
    Edit: I just realized in my second last step before dividing both sides by w2 I have the answer, is it because they mention that the two oscillators are identical and that's why the value for w does't matter?
     
  6. Apr 1, 2015 #5

    BvU

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    "Identical oscillators" means they have the same ##\omega##.
     
  7. Apr 1, 2015 #6

    rude man

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    That can't be acceleration; the units are wrong.
    The answer given to you makes no sense: initial conditions are dx/dt(0+) and d2x/dt2(0+), not a quantity with dimension L2T-2.
    I think BvU has the intended interpretation of this confused problem statement right.
    If so, the question should have been worded as follows: "what function (combination) of x(0) and v(0) must be constant for an undamped harmonic oscillator to exhibit the same amplitude of oscillation?"
    In other words there was never a need to mention two oscillators at all.
     
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