1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Simple harmonic oscillator

  1. Nov 6, 2009 #1
    1. The problem statement, all variables and given/known data

    Show simple harmonic motion starting from Hooke's Law.

    3. The attempt at a solution





    [tex]\int\int\frac{1}{x}d\left(dx\right)=-\frac{k}{m}\int dt\int dt[/tex]




    But it should be:


    I thought I knew how to do this. :redface:
  2. jcsd
  3. Nov 6, 2009 #2


    User Avatar
    Homework Helper

    Well normally you could just put F=ma and then put it the form [itex]a= - \omega^2 x[/itex] and that would be sufficient to show SHM.

    Else to solve the DE, you would need to know that the equation y''+ky=0 has solutions y1=cos(kt) and y2=sin(kt)
  4. Nov 7, 2009 #3
    I want to solve the differential equation. But as you can see, somewhere I need to take the square root of the exponential argument.

    Where, in my steps, did I miss that?
  5. Nov 7, 2009 #4
    Nobody knows?
  6. Nov 7, 2009 #5
    I don't see how to use this to get to the right answer, but:

    [tex]\int\frac{1}{x}dx \neq \ln (x)[/tex]

    [tex]\int\frac{1}{x}dx = \ln |x| + C[/tex]

    So the derivation would continue:

    [tex]\ln |x| + C =-\frac{k}{2m}t^2 [/tex]

    [tex] |x| = e^{-\frac{k}{2m}t^2 + C} [/tex]

    Like I said, not sure how to use this, or if it helps. I could see squaring x, then taking the square root, in lieu of the absolute value signs, but, then, I don't see where the i comes out. Most of the derivations I've seen 'guess' at the solution of the 2nd order DE to be of the form [itex]a cos(\omega t)[/itex], and go from there.

    I'm going to look at this more tomorrow, I can't believe I don't know either.
  7. Nov 7, 2009 #6


    User Avatar
    Homework Helper

    [tex]m\frac{d^2x}{dt^2}=-kx \Rightarrow \frac{d^2x}{dt^2}+\frac{k}{m}x=0 [/tex]

    the auxiliary equation is r2+(k/m)=0 so r=±√(k/m)i

    when you have roots in the form r=λ±μi what is the general solution x(t) equal to?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook