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Choosing unit vectors for harmonic motion problems

  1. Jan 15, 2015 #1
    Consider a vertical pendulum affected by gravity (See the pdf file i included). Now i can choose two different opposite directions for my unit vectors which give me different equations.
    [tex]\downarrow : m\ddot x = mg-kx [/tex]
    [tex]\uparrow : m\ddot x = kx-mg[/tex]
    Which of course makes perfect sense, changing direction changes the sign. The problem is now if i want to solve them the second case yields a weird result.

    so in the first case the (real) solution would be (if we set ##\omega _n^2 = \frac{k}{m}##)
    [tex]x = Acos(\omega _n t )+ Bsin(\omega _n t) + \frac{mg}{k}[/tex]
    and for the second case
    [tex]x = Ae^{\omega _n t} + Be^{-\omega _n t} +\frac{mg}{k}[/tex]

    So what I'm wondering why i would get a different solution just by changing the direction of the unit vector and how i can reconcile the approaches or know how i should choose the direction of my unit vectors.

    Attached Files:

  2. jcsd
  3. Jan 15, 2015 #2


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    Whatever your direction for x, the force of the spring is always in the opposite direction to the displacement. So, in the second case, you should have:

    [tex]\uparrow : m\ddot x = -kx-mg[/tex]
  4. Jan 15, 2015 #3
    Thank you, I understand now! You won't believe I've been thinking about this for several hours before i wrote this :)
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