Question

A thin insulating thread of length L is attached to a point on the perimeter of a thin disk.
Disk is has radius R< L and is charged with a uniform surface density, σ>0. At the end of
the thread there is a small ball mass m and charge q>0. The mass happens to be located
on disk's axis. What is the mass of the ball?

Answer 1

Figure the Electric field from the charged disk along the axis at a distance z, given R

http://hyperphysics.phy-astr.gsu.edu/HBASE/electric/elelin.html#c3

Now, given that expression, then you know from the geometry with the angle theta the string makes from disk,
tanTheta=Eq/mg
But CosineTheta=R/L
and sinTheta=sqrt((L^2-R^2)/L^2) or
sinTheta=sqrt(1-(R/L)^2)
but sinTheta/cosTheta= tan Theta, so

Eq/mg=sqrt(1-(R/L)^2) *L/R
Eq/mg=sqrt((L/R)^2-1)
and now you can solve for L

Remember in your E expression, z=Sqrt(L^2-R^2)

As you can tell, some algebra will be required. Get a nice pad of paper.