A planar loop of wire with total electrical resistance \(R\) and area \(A\) rotates at a constant angular speed \(\omega\) in a region of uniform, constant magnetic field \(\vec{B}\). The rotation axis lies in the plane of the loop and is perpendicular to the magnetic field. If the angular speed of the loop is increased from an initial value \(\omega_0\) to \(2\omega_0\), by what factor does the total thermal energy dissipated in the loop during one complete revolution change?