Force on Steel Bar Inside Solenoid

[abilolado]

Hello all.
Me and some friends are building a coil cannon, and we've been doing some calculations [I know its unecessary but... well, we're physicists! (well, physics students...)]. But we got stuck.
How to calculate the force acting on a steel bar (or some other ferromagnetic material, maybe there's a constant related to each type of material for this application) of length $L_{bar}$ and radius $R_{bar}$ inside a solenoid of $N$ turns with current $i$ flowing through it ($R_{bar}<R_{solenoid}$)?
I found approximations that the magnetic field inside the solenoid is $B=\mu N i$ (using the right hand rule to find the direction, no problem there). I don't know, however, how to get the force applied on a steel bar under this field. Even an approximation would be great (But exact solutions are very appreciated, even more if they have the whole process of deriving the equation, hehe).
From the demonstrations online, I see that the bar oscillates going in and out of the coil. So I'm guessing there's some sort of harmonic oscillation approximation.
Much like shown in this clip (2:13):



(Unless the damping comes from something other then friction, there's no need to include it)

PS: Since we're using a capacitor bank, the current will not be constant, but I guess I can integrate the approximation over a varying current anyway.

 

[berkeman]

How big is your capacitor bank? What size projectile, and with what planned exit velocity?

 

[abilolado]

How big is your capacitor bank? What size projectile, and with what planned exit velocity?

Everything is arbitrary for now.

 

[berkeman]

Everything is arbitrary for now.

Whelp, if you are hoping to work with many kJ of capacitively stored energy and fire medium size projectiles at many meters per second, this thread will be closed as dangerous. If you are looking to make a small table-top demonstration with 12V capacitive storage of a little bit of energy, that may be okay.

BTW, I didn't watch the video. Solenoids work by pulling ferrous metal bars into the coil. How do you make an ejector mechanism out of that?

 

[abilolado]

Perhaps I'm misunderstanding the meaning of a solenoid. The whole point is to have the projectile, or whatever it is, being accelerated due to the magnetic field produced by the current in a looped wire. The current would be cut off once the projectile reached the middle of the coil so that there be no force pulling it back in, like in the video.
I just need to know the equations describing the forces or the displacement so we can make predictions about the final velocity.
We're just a physics club under the supervision of the university, there's not much we can to do make it dangerous in any way.

 

[Vanadium 50]

We're just a physics club under the supervision of the university,

And nobody at this university can help you with these calculations? How is this "supervision"?

there's not much we can to do make it dangerous in any way.

That's absurd. "We don't know what we are doing so we can't make it dangerous" is simply untrue.