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Suppose we have a pendulum which is fixed at some point, P. We release it from some initial angle $\theta$ and it swings over a solenoid for which there exists some magnetic force. Note that there is a piece of string connecting from one end at a fixed point P, and the mass being swung is a small magnet at the other end.

Apparatus

If I wish to model this pendulum's motion, I believe that I cannot use the generic formula; $$y(t)=Ae^{-kt}cos(\omega t)$$ I believe that a more refined model may be required which includes the effect of such a magnetic field on the mass.

However, deriving this is seeming to be very difficult for me... Is there a possible way where we can include the effect of the magnetic field on the original mechanical system?

I have attempted to make an ODE in such a way that; $$y''+\alpha y'+\beta y=0$$ Where $\alpha$ and $\beta$ are just constants caused by friction.
After further thought I have attempted to make an ODE by considering the equation;

$$I(\omega^2)''=\tau_{gravity}+\tau_{Dampening}+\tau_{Magnetic} $$

Would this be on the right track? Or is there another path which I need to take?

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  • $\begingroup$ Have you established what the force between magnet and solenoid is? And what are the $\sigma$ representing? $\endgroup$
    – Gert
    Commented Sep 6, 2019 at 13:35
  • $\begingroup$ You cannot, for reasonable definitions. But I don't think that's what you mean. I think you mean "some simple formula like this"? If so, consider standard treatments generally ignore friction for clarity. $\endgroup$ Commented Sep 6, 2019 at 15:23
  • $\begingroup$ I'm more concerned about adding another term to the original ODE. But I don't know what the term should be... That way I account for the magnetic field strength. $\endgroup$
    – JayP
    Commented Sep 6, 2019 at 22:38

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