Work-Energy For Rotation And Rolling

A wheel with mass $M$, radius $R$, and moment of inertia $I$ is brought to rest by a constant frictional torque $\tau$ at its axle in two scenarios. In the first, the wheel rotates about a fixed axle with an initial angular velocity of $\omega_0$ and stops after covering an arc length of $d_{\text{rotation}}$. In the second, the wheel rolls without slipping with an initial speed of $v_0 = \omega_0 R$ and stops after traveling a distance of $d_{\text{rolling}}$. Which equation correctly relates $d_{\text{rolling}}$ and $d_{\text{rotation}}$?