Conservation of angular momentum exercise

Exercise:

A disk of radius $R$ and moment of inertia $I_1$ rotates with angular velocity $\omega_0$. The axis of a second disk, of radius $r$ and moment of inertia $I_2$ is at rest. The axes of the two disks are parallel. The disks are moved together so that they touch. After some initial slipping the two disks rotate together. Find the final rate of rotation of the smaller disk.

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Attempt:

$L_{1_0} = L_1 + L_2 \rightarrow I_1\omega_0 = I_1\omega_1 + I_2\omega_2$

$\omega = \frac{v}{r} \rightarrow v = \omega r$

$\omega_1 R = \omega_2 r \rightarrow \omega_1 = \frac{r}{R}\omega_2$

$I_1\omega_0 = I_1\frac{r}{R}\omega_2 + I_2\omega_2 \rightarrow \omega_2 = \frac{I_1\omega_0}{\frac{r}{R}I_1 + I_2}$

$$\omega_2 = \frac{I_1\omega_0}{\frac{r}{R}I_1 + I_2}$$

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Request:

Is my solution correct? If not, where and why?

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