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Period of an Oscillating Particle

  1. Sep 24, 2013 #1
    1. The problem statement, all variables and given/known data

    A particle of mass m is oscillating with amplitude A in 1D (without damping). Using conservation of energy, I'm asked to determine the period T with the correct proportionality factor and period of integration.

    2. Relevant equations

    Potential energy: [itex] U(x) = a x^2 + b x^4 [/itex] where a > 0 and b ≥ 0
    Period: [itex]T(A) \propto[/itex]∫[itex]\frac{dx}{\sqrt{E(A) + U(X)}}[/itex] (from conservation of energy)

    3. The attempt at a solution

    I scribbled down a few attempts but none of them went anywhere. I don't have the slightest idea where to start so I'm hoping someone can point me in the right direction.

    I think that's all the information provided, but feel free to ask if you have any questions. It's my first time posting here, my apologies if I haven't follow proper rules or etiquette.

    Thanks
     
  2. jcsd
  3. Sep 24, 2013 #2

    Simon Bridge

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    Welcome to PF;
    I take it the oscillations are harmonic and though undamped, there may be a driving force?
    If it were SHM, then the potential energy terms would be quadratic right?

    It looks like you are expected to exploit the relationship you already know about between kinetic and potential energy. i.e. you need to examine what else you know about energy in oscillations - which should net you an appropriate proportionality and a region of integration.

    Compare with how it's done for SHM.
     
  4. Sep 25, 2013 #3
    That appears to be the case. I found some relevant relations while looking through the notes again.

    [itex]F = -\frac{dU}{dx} [/itex]

    So that would give a driving force of

    [itex]F = -2ax -4b x^3 = m\ddot{x} [/itex]

    The Hamiltonian is also given as

    [itex]H = \frac{p^2}{2m} + U(x) [/itex]

    Would that be interchangeable with the total energy E(A)? In that case I would be able to integrate using just the kinetic energy.
     
  5. Sep 26, 2013 #4

    Simon Bridge

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    Is the Hamiltonian interchangeable with the total energy?
    Do you know the relationship between kinetic and potential energy for your oscillator?

    Wouldn't F=-dU/dx be the restoring force rather than the "driving force"?
    The problem statement only says that the oscillation is un-damped, does not say there is no driving force - I don't know, there may be: is there?
     
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