The stored energy in the spring is 0.5*k*dx^2. This energy is transferred to the kinetic energy of the mass and rotational energy of the turntable. If v is the mass velocity on release, its kinetic energy is
The rotational energy of the turntable is
0.5*I*w^2, where w = angular velocity of the turntable.
The sum of these must equal the initial energy in the spring.
You then use conservation of angular momentum to get another relation. (Consider angular momentum about the center of the turntable.) The angular momentum of the mass on release is m*v*b; the angular momentum of the turntable is w*I. Since the initial momentum of the system was zero, these must be equal and opposite, so
w*I = -m*v*b; .........this with the energy equivalence
.5*m*v^2 + 0.5*I*w^2 = 0.5*k*dx^2
gives you two equations in two unknowns, w and v.
solve for v in terms of w from the first, plug into the second to get w, then put that back into the first to solve for v.
The rotational speed will come out in radians per second; convert to rpm using one rev = 2π radians, one minute = 60 seconds. rpm = (30/π)*w