Problem about uniform thin disk

[Another]

Why do they only think of kinetic energy of motion?
Why don't they think of both kinetic of motion and kinetic of rolling energy?

So. i think
$L = \frac{1}{2}mv^2 + \frac{1}{2}I \omega^2$
$L = \frac{1}{2}m(r \omega)^2 + \frac{1}{2}(\frac{1}{2}mr^2) \omega^2$
$L = \frac{3}{4} mr^2 \omega^2$

 

[kuruman]

Look at the last three equations in the answer. The result will be the same whether you multiply $mr^2\omega^2$ by $\frac{3}{4}$ or $\frac{1}{2}$. It looks like whoever wrote the solution knew the bottom line and became sloppy.