I have a horizontal bar on vertical rails and we can consider that does not exist any friction.

Perpendicular to the plane of the rails it exists a magnetic field $B$.

I am asked to find the movement equations and the bar position depending on time.

And I have made some considerations:

I should separate the phenomenon in two parts: first one until the forces are equal and one after.

I'm stuck on finding the equations before the forces get equals. This is what I have done so far:

$$mg-F_{m}=m·a$$ with $F_m$ the magnetic force, with the Ohm Law: $$\epsilon=\frac{I}{R}$$ and with: $$\epsilon=-\frac{d \phi }{dt}=-Bl\frac{dx}{dt}$$ and all together:

$$mg-\frac{B^2l^2}{R}·\frac{dx}{dt}=m·\frac{dx^2}{dt^2} $$

So, what now? I am in my first physics course so I am not supposed to be solving this by two order differential equations tools.

I know that I can find the $v_{limit}$ if I make the sum of forces equal to $0$, but that is not enough. I'm really stuck, maybe I should be making some assumptions, but I don't know what!

(I have tried my best with my English language knowledge)

Thank you in advance.


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