Please take a look at number b. Maybe I will need help in solving this. Let me try first. But what do they even want here? What does the critical velocity mean? And is B (beta) any constant here, or is there any standard value for B?

I don't know what they mean by "critical" velocity, but would guess that they want you to derive a formula for the velocity of the sphere. As charge leaks away the angle theta will decrease and the sphere will swing downward toward the plate - derive a formula for the velocity of the sphere as a function of time, in terms of q_0, m, and beta..

Beta is merely a constant; it does not have a particular value.

Thanks for the explanation. :) now let me try my hand at this question.