Question $8\mathrm{.}3$

Two identical cylindrical rollers are installed onto fixed rotation axes aligned parallel within a horizontal plane. The distance between the axes is $2l.$ A uniform board of mass $m$ is placed onto the rollers. The coefficient of kinetic friction between the rollers and the board is $.$ Suppose that the centre of mass of the board is displaced from the central position between the rollers by a small distance of $x.($a$)$ Find the normal force acting on the board by each cylindrical roller.

$($b$)$ The rollers are driven by external rotors to rotate at uniform angular speeds in directions shown in the figure. Show that the motion of the board is a harmonic oscillation and find its period.