I tried something else, and I think I am a little closer to a solution.
τ = Fdsinθ, τ = Iα, α = a/r, d = r = the radius of the spool, F = mhg - mha
Fr = Iα
Fr = I(a/r)
Fr2 = Ia
(Fr2/a) = I
I = [r2(mhg - mha)]/a
This solution is closer, but instead of 'α' in the denominator, there is just...