Just a kinematics problem from a Russian physics olympiad

[wrobel]

Perhaps someone would be interested in offering it to their students.

There is a disk of radius $R$. The disk is wound with two inextensible strings, the ends of which are attached to the ceiling. At the moment in time shown in the picture, the strings are taut and the angle between them is equal to $\alpha$. The disk has an angular velocity $\omega$. Find the velocity of the disk's center.

 

[anuttarasammyak]

Spoiler

$$(\mathbf{v}-R\omega\mathbf{e_1})\cdot\mathbf{e}_1=0$$
$$(\mathbf{v}-R\omega\mathbf{e_2})\cdot\mathbf{e}_2=0$$
So
$$\mathbf{v}\cdot\mathbf{e_1}=\mathbf{v}\cdot\mathbf{e_2}=R\omega$$
and
$$\mathbf{e_1}\cdot\mathbf{e_2}=\cos\alpha$$
Thus
$$\mathbf{v}=\frac{R\omega}{\cos\frac{\alpha}{2}} \frac{\mathbf{e_1}+\mathbf{e_2}}{2\cos\frac{\alpha}{2}}$$
where obviously
$$|\frac{\mathbf{e_1}+\mathbf{e_2}}{2\cos\frac{\alpha}{2}}|=1$$