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Main Question or Discussion Point
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 = mgkx [/tex]
[tex]\uparrow : m\ddot x = kxmg[/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.
[tex]\downarrow : m\ddot x = mgkx [/tex]
[tex]\uparrow : m\ddot x = kxmg[/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.
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