A uniform solid cylinder of mass  $\text{'}m\text{'}$  rests on two horizontal planks. A light thread is wound on the cylinder. The hanging end of the thread is pulled vertically down with a constant force $F,$  which still does not bring any sliding of the cylinder. The coefficient of friction between the cylinder and planks is equal to  $\mu .$ If the maximum acceleration $a_{\max }(inm/s^2)$  of the axis of cylinder rolling over the planks is $a_0,$  then what is the value of  $\frac{3}{4}{a}_0?$ (Assume $\mu =1/3$ and take  $g=10m/{\mathrm{s}}^2$)