[tex]N_1m_1=N_2m_2[/tex] Where everything subscripted 1 is on one side and everything 2 is on the other, N is force, and m is distance away from the fulcrum. Rearranging it produces [tex]N_1\frac{m_1}{m_2}=N_2[/tex]. This only useful for equilibrium problems, i.e. the arm is balanced and not moving. Removing the force on the one side, will only allow the force of the other side to move down, i.e. no force is tranfered.
Not sure what you mean by your wording and the fact that your example is a non-equilibrium problem. That sounds more like a catapult, in which case you would be more interested in energy, momentum, and angular momentum of the arm so you know how massive to make your base so it doesn't jump up and crush someone or something important.