Results 1 to 8 of 8

Math Help - [SOLVED] Simple Harmonic Motion

  1. #1
    Junior Member Evales's Avatar
    Joined
    Oct 2007
    Posts
    63

    [SOLVED] Simple Harmonic Motion

    How do you prove that a function shows that something is in SHM if the equation isn't in the form:

    x = Acos (kt + b)

    Is a graph would be acceptable or is there a mathematical way to do this?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by Evales View Post
    How do you prove that a function shows that something is in SHM if the equation isn't in the form:

    x = Acos (kt + b)

    Is a graph would be acceptable or is there a mathematical way to do this?
    Post the function.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member Evales's Avatar
    Joined
    Oct 2007
    Posts
    63
    x = sin t + [(3)^(1/2)]cos t
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,888
    Thanks
    326
    Awards
    1
    Quote Originally Posted by Evales View Post
    How do you prove that a function shows that something is in SHM if the equation isn't in the form:

    x = Acos (kt + b)

    Is a graph would be acceptable or is there a mathematical way to do this?
    (Edited version)

    The equation you have written is one way to define harmonic motion. So if your trial function cannot be put into this form, then it does not represent a SHO.

    On the other hand you can define simple harmonic motion as x"(t) = -Ax(t). By taking small displacements from equilibrium for many physical systems you will find it is a SHO. (Usually you need to do a Taylor expansion to prove this. The linear coefficients in the series needs to be 0 and the quadratic coefficient needs to be negative. The condition on the linear term is automatic around an equilibrium point, so many systems have this kind of behavior.)

    For your function
    sin(t) + A~cos(t)
    you can easily see that it is the (linear) sum of two SHO functions, so it is also a harmonic oscillator.

    -Dan
    Last edited by topsquark; May 31st 2008 at 08:03 AM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by topsquark View Post
    For your function
    sin(t) + A~cos(t)
    you can easily see that it cannot be put into cos(kt + b) form.
    You might easily be able to see that it cannot be put in that form, but I can easily see that it can .

    RonL
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member Evales's Avatar
    Joined
    Oct 2007
    Posts
    63
    Quote Originally Posted by CaptainBlack View Post
    You might easily be able to see that it cannot be put in that form, but I can easily see that it can .

    RonL
    Any way of enlightening me?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,888
    Thanks
    326
    Awards
    1
    Quote Originally Posted by CaptainBlack View Post
    You might easily be able to see that it cannot be put in that form, but I can easily see that it can .

    RonL
    Yeah, I was just coming back to fix that.

    -Dan
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,888
    Thanks
    326
    Awards
    1
    Quote Originally Posted by Evales View Post
    Any way of enlightening me?
    x = sin(t) + A~cos(t)

    x' = cos(t) - A~sin(t)

    x" = -sin(t) - A~cos(t) = -x

    So this is a harmonic oscillator.

    Or
    sin(t) + A~cos(t) = \sqrt{1 + A^2}~sin(t + \phi)
    (where \phi is a constant) which is a sine function and hence a harmonic oscillator.

    (This is from the more general:
    a~sin(t) + b~cos(t) = \sqrt{a^2 + b^2}~sin(x + \phi)
    where
    \phi = tan^{-1} \left ( \frac{b}{a} \right )
    for a \geq 0
    and
    \phi = tan^{-1} \left ( \frac{b}{a} \right ) + \pi
    for a < 0)

    -Dan
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Simple Harmonic Motion
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 10th 2010, 05:18 AM
  2. Simple Harmonic Motion
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 2nd 2010, 04:22 AM
  3. Simple Harmonic Motion
    Posted in the Trigonometry Forum
    Replies: 3
    Last Post: May 16th 2009, 05:14 AM
  4. Simple Harmonic Motion
    Posted in the Advanced Applied Math Forum
    Replies: 7
    Last Post: June 2nd 2008, 04:27 AM
  5. simple harmonic motion
    Posted in the Advanced Applied Math Forum
    Replies: 10
    Last Post: January 4th 2008, 12:09 PM

Search Tags


/mathhelpforum @mathhelpforum