Steady state

A homogeneous plank of mass $M$ and length $L$ is released gently on an inclined rolling mill, which consists of a large number of uniform horizontal cylindrical rollers. Axes of the roller are fixed parallel to each other in a plane inclined at an angle $\theta$ with the horizontal with a separation $l(l<< L)$ between two adjacent axes. The mass of a roller is $m$ , each roller can rotate about its axis without friction and acceleration due to gravity is $g$ .Long time after the plank is released; it acquires a steady speed $v$ . Find this steady speed.

Answer comes in the form of $v=\alpha \sqrt{\frac{Mgl \sin \theta}{m}}$
Type $\alpha=?$