A motor that exerts a constant torque of magnitude \(\tau\) spins up two flywheels from rest about their central axes. One is a uniform solid disk with rotational inertia \(\dfrac{1}{2}MR^2\), and the other is a thin hoop with rotational inertia \(MR^2\). Both flywheels have mass \(M\) and radius \(R\), and frictional torques are negligible. The motor acts on the disk for a time interval \(t\) and on the hoop for a time interval \(2t\). What is the ratio \(\dfrac{\omega_{\text{disk}}}{\omega_{\text{hoop}}}\) of the disk's final angular speed to the hoop's final angular speed?