Conservation of angular momentum

Question:
A solid cylinder with a moment of inertia, $I_0$, rotates about its center at an angular velocity of $\omega_0$. Another solid cylinder, which is initially rest, is put onto the rotating one gently. Both eventually rotate at an angular velocity, $\omega_f$. Find the velocity $\omega_f$.

Answer:
The angular momentum is defined as
\[
L = I\omega
\]
With the constant velocity, initial and final angular momenta are conserved. Namely, initial total momentum = final total momentum. In this case, we have
\[
I_0 \omega_0 = (I_0 + I_1)\omega_1
\]
Solve for $\omega_f$.
\[
\omega_f = \frac{I_0 \omega_0}{I_0 + I_1}
\]