Question

During the experiment if you could double the breakaway magnetic force with all other quantities left unchanged, what is the new value for the critical velocity if it was vo initially?

Answer 1

There are some missing information in the question.
However, since you are talking about magnetic force, I think you refer to the Lorentz force. When a particle of charge q and velocity v is immersed in a magnetic field of intensity B, the force acting on the particle is:
$F=qvBsin\theta$
where $\theta$ is the angle between the magnetic field and the direction of the particle.
Therefore, if force F is doubled, then also the velocity v must be double of its initial value:
$v=2v_0$