From my limited knowledge on the subject I tried to solve a simpler problem  a point mass undergoing circular motion about an axis. Let a constant torque(T) starts acting at t=0, the equation of motion will be
dL/dt = T, here L is the angular momentum.
Integrating both the sides we find
L = Tt
Or Iw = Tt { I is the moment of inertia of the particle(=MR^2) and w,the angular velocity}
Or MR^2w = Tt
Or Mo^2R^2w/sqrt(1v^2/c^2) = Tt, {I put M=Mo/sqrt(1v^2/c^2}
Replacing Rw for v and solving, I get
w = T^2t^2c/sqrt(c^2*Mo^2*R^4 + T^2*t^2*R^2).
I am not sure about this. Please check.
Thanks.
