1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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]
    Define:
    [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

    tiny-tim

    User Avatar
    Science Advisor
    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

    Hootenanny

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Simple Harmonic Oscillator Help
Loading...