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Homework Help: Harmonic Motion and Springs Question

  1. Jul 22, 2006 #1
    I'm trying to derive the formula for the period T as a function of the mass of the object. Here is my attempt. Note that I cheated and passed the section I had trouble with, without fully understanding it.

    http://www.filehive.com/files/0722/image.jpg

    Much of this is quite straightforward for me.

    [tex]y = A sin \alpha[/tex] (basic trigonometry)
    ----
    Here is the part I'm not sure I understand fully:

    [tex]\alpha = \omega t[/tex]

    Why does that equation work?
    -----

    The angle alpha is replaced by [tex]\psi t[/tex] so the result is:

    [tex]y = A sin \omega t[/tex]

    The speed in the y direction is as follows:

    [tex]v(t) = \frac {dy} {dt} = \omega A cos \omega t[/tex]

    The acceleration in the y direction is:

    [tex]a(t) = \frac {d^2x} {dt^2} = -\omega^2 A sin \omega t[/tex]

    The above is simple calculus.

    When a net force is acting on a body an acceleration will show.

    The force that is acting on the body is (via hookes law):

    [tex]F = -ky[/tex]

    If we replace y with the expression we derived earlier we get

    [tex]F = ma = -m \omega^2 A sin \alpha t[/tex]
    [tex]F = -ky = -k A sin \alpha t[/tex]

    ie.

    [tex]m \omega^2 = -k \Leftrightarrow \omega = \sqrt{k / m}[/tex]

    Combining the above expression with the commonly known

    [tex]\omega = 2 \pi / T[/tex]

    and you get the final result

    [tex]T = 2 \pi \sqrt {m / k}[/tex]

    ------

    My question is:

    Why is [tex]\alpha = \omega t[/tex]?

    Thank you for your time. Have a nice day.
     
    Last edited: Jul 22, 2006
  2. jcsd
  3. Jul 22, 2006 #2

    arildno

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    They assume the angular velocity is a constant.
    It probably isn't, so they've hidden away an order of magnitude argument that would show it is a good approximation.

    (That is, they've hidden away everything that physics is about)
     
  4. Jul 22, 2006 #3
    So it is basically because

    [tex]\alpha = \omega t[/tex]

    Unit:

    rad = rad/s x s

    with the approximation that angular velocity is constant?
     
  5. Jul 22, 2006 #4

    arildno

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    Yep, that should be it.
     
  6. Jul 22, 2006 #5
    I was just wondering why you used psi instead of omega. Nothing wrong with it, but it's not written that way, usually.

    alpaha = omega.t is just the rotational counterpart of s = vt (linear motion with constant velocity).
     
  7. Jul 22, 2006 #6
    Yes, I noticed that, so I changed it. The "how-to-latex" got me confused for a bit before i realised it.

    Thank you arildno and neutrino! I really should check the units (as in rad = rad/s x s) more often :smile:
     
  8. Jul 22, 2006 #7

    nrqed

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    The question is not explicitly provided but I was under the impression that that goal was to relate the motion of a mass attached to an ideal spring to circular motion. In that case, using a constant angular velocity is not an approximation or a guess. It follows from the fact that the projection along one of the axis must represent simple harmonic motion. And that implies a constant omega.

    Just a comment.

    Regards

    Patrick
     
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