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Homework Help: Simple Harmonic Oscillator Help

  1. Jul 16, 2008 #1
    1. The problem statement, all variables and given/known data

    A particle oscillates between the points x = 40mm and x = 160mm with an acceleration a = k(100-x) where k is a constant. The velocity of the particle is 18mm/s when x=100 and zero at x = 40mm and x = 160mm. Determine a) the value of hte constant k, b) the velocity when x = 120mm

    2. Relevant equations

    [tex]a = k(100-x)[/tex]

    3. The attempt at a solution

    This looked like a simple harmonic oscillator to me.

    So I went:

    [tex]a = 100k - kx [/tex]

    [tex]\frac{d^2x}{dt^2} = 100k - kx[/tex]
    [tex]\dot x = \frac{\mathrm{d}x}{\mathrm{d}t}[/tex]
    Then Observe:
    [tex]\frac{\mathrm{d}^2 x}{\mathrm{d} t^2} = \ddot x = \frac{\mathrm{d}\dot {x}}{\mathrm{d}t}\frac{\mathrm{d}x}{\mathrm{d}x}=\frac{\mathrm{d}\dot {x}}{\mathrm{d}x}\frac{\mathrm{d}x}{\mathrm{d}t}=\frac{\mathrm{d}\dot{x}}{\mathrm{d}x}\dot {x}[/tex]
    Then substitute:
    [tex]\frac{d\dot x}{dx}\dot x = 100k-kx [/tex]

    [tex]d\dot x = (100k-kx)dx [/tex]

    [tex]\int \dot x d\dot x = \int (100k-kx)dx [/tex]

    [tex]\dot x^2 = 50kx - kx^2 + c[/tex]

    I got that far in the manipulation, then I got stuck. Where do i go from here or what have I done wrong? My current approach is to solve for the differential then differentiate to get an equation for the velocity. Is there a better approach?
    Last edited: Jul 16, 2008
  2. jcsd
  3. Jul 16, 2008 #2


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    Homework Helper

    Welcome to PF!

    Hi noodle_snacks! Welcome to PF! :smile:

    hmm … a bit long-winded …

    I'd have started by saying "Let y = x - 100"

    Then that gives you y'' = -ky, which you may be able to solve on sight.

    If not, then continue y''y' = -kyy', and so on.

    It isn't any better … but it is easier! :biggrin:

    You got stuck at:
    So square-root it, and you get dx/√(....) = constant, and you can use trigonometric substitution to solve that. :smile:
    Last edited: Jul 16, 2008
  4. Jul 16, 2008 #3


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    Re: Welcome to PF!

    Just to point out here that there is no need to actually solve your final differential equation. The question only asks you to determine the value of k and the value of the velocity for a given displacement, both of which can be done by just plugging numbers into the ODE without actually solving it.
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